(International Tables) Reciprocal space

International Tables for Crystallography
Volume B: Reciprocal space

First online edition (2006) ISBN: 978-0-7923-6592-1 doi: 10.1107/97809553602060000102 | 1 | 2 |

Edited by U. Shmueli

Preface to the second edition (pp. xxv-xxvi) | html | pdf |
U. Shmueli

General relationships and techniques
- 1.1. Reciprocal space in crystallography (pp. 2-9) | html | pdf | chapter contents |
  U. Shmueli
  - 1.1.1. Introduction (p. 2) | html | pdf |
  - 1.1.2. Reciprocal lattice in crystallography (pp. 2-3) | html | pdf |
  - 1.1.3. Fundamental relationships (pp. 3-4) | html | pdf |
    - 1.1.3.1. Basis vectors (p. 3) | html | pdf |
    - 1.1.3.2. Volumes (pp. 3-4) | html | pdf |
    - 1.1.3.3. Angular relationships (p. 4) | html | pdf |
    - 1.1.3.4. Matrices of metric tensors (p. 4) | html | pdf |
  - 1.1.4. Tensor-algebraic formulation (pp. 5-7) | html | pdf |
    - 1.1.4.1. Conventions (p. 5) | html | pdf |
    - 1.1.4.2. Transformations (p. 5) | html | pdf |
    - 1.1.4.3. Scalar products (pp. 5-6) | html | pdf |
    - 1.1.4.4. Examples (pp. 6-7) | html | pdf |
  - 1.1.5. Transformations (pp. 7-8) | html | pdf |
    - 1.1.5.1. Transformations of coordinates (pp. 7-8) | html | pdf |
    - 1.1.5.2. Example (p. 8) | html | pdf |
  - 1.1.6. Some analytical aspects of the reciprocal space (pp. 8-9) | html | pdf |
    - 1.1.6.1. Continuous Fourier transform (p. 8) | html | pdf |
    - 1.1.6.2. Discrete Fourier transform (pp. 8-9) | html | pdf |
    - 1.1.6.3. Bloch's theorem (p. 9) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 1.1.4.1. Derivation of the general expression for the rotation operator (p. 6) | html | pdf |
- 1.2. The structure factor (pp. 10-24) | html | pdf | chapter contents |
  P. Coppens
  - 1.2.1. Introduction (p. 10) | html | pdf |
  - 1.2.2. General scattering expression for X-rays (p. 10) | html | pdf |
  - 1.2.3. Scattering by a crystal: definition of a structure factor (p. 10) | html | pdf |
  - 1.2.4. The isolated-atom approximation in X-ray diffraction (pp. 10-11) | html | pdf |
  - 1.2.5. Scattering of thermal neutrons (p. 11) | html | pdf |
    - 1.2.5.1. Nuclear scattering (p. 11) | html | pdf |
    - 1.2.5.2. Magnetic scattering (p. 11) | html | pdf |
  - 1.2.6. Effect of bonding on the atomic electron density within the spherical-atom approximation: the kappa formalism (pp. 11-13) | html | pdf |
  - 1.2.7. Beyond the spherical-atom description: the atom-centred spherical harmonic expansion (pp. 14-15) | html | pdf |
    - 1.2.7.1. Direct-space description of aspherical atoms (pp. 14-15) | html | pdf |
    - 1.2.7.2. Reciprocal-space description of aspherical atoms (p. 15) | html | pdf |
  - 1.2.8. Fourier transform of orbital products (pp. 17-18) | html | pdf |
    - 1.2.8.1. One-centre orbital products (p. 18) | html | pdf |
    - 1.2.8.2. Two-centre orbital products (p. 18) | html | pdf |
  - 1.2.9. The atomic temperature factor (p. 18) | html | pdf |
  - 1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation (pp. 18-19) | html | pdf |
  - 1.2.11. Rigid-body analysis (pp. 19-22) | html | pdf |
  - 1.2.12. Treatment of anharmonicity (pp. 22-23) | html | pdf |
    - 1.2.12.1. The Gram–Charlier expansion (p. 22) | html | pdf |
    - 1.2.12.2. The cumulant expansion (pp. 22-23) | html | pdf |
    - 1.2.12.3. The one-particle potential (OPP) model (p. 23) | html | pdf |
    - 1.2.12.4. Relative merits of the three expansions (p. 23) | html | pdf |
  - 1.2.13. The generalized structure factor (pp. 23-24) | html | pdf |
  - 1.2.14. Conclusion (p. 24) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 1.2.7.1. Real spherical harmonic functions (x, y, z are direction cosines) (pp. 12-13) | html | pdf |
    - Table 1.2.7.2. Index-picking rules of site-symmetric spherical harmonics (Kara & Kurki-Suonio, 1981) (p. 15) | html | pdf |
    - Table 1.2.7.3. `Kubic Harmonic' functions (pp. 16-17) | html | pdf |
    - Table 1.2.7.4. Closed-form expressions for Fourier transform of Slater-type functions (Avery & Watson, 1977; Su & Coppens, 1990) (p. 19) | html | pdf |
    - Table 1.2.8.1. Products of complex spherical harmonics as defined by equation (1.2.7.2a) (p. 20) | html | pdf |
    - Table 1.2.8.2. Products of real spherical harmonics as defined by equations (1.2.7.2b) and (1.2.7.2c) (p. 20) | html | pdf |
    - Table 1.2.8.3. Products of two real spherical harmonic functions $[y_{lmp}]$ in terms of the density functions $[d_{lmp}]$ defined by equation (1.2.7.3b) (p. 21) | html | pdf |
    - Table 1.2.11.1. The arrays $[G_{ijkl}]$ and $[H_{ijkl}]$ to be used in the observational equations $[U_{ij} = G_{ijkl} L_{kl} + H_{ijkl} S_{kl} + T_{ij}]$ [equation (1.2.11.9)] (p. 21) | html | pdf |
    - Table 1.2.12.1. Some Hermite polynomials (Johnson & Levy, 1974; Zucker & Schulz, 1982) (p. 22) | html | pdf |
- 1.3. Fourier transforms in crystallography: theory, algorithms and applications (pp. 25-98) | html | pdf | chapter contents |
  G. Bricogne
  - 1.3.1. General introduction (p. 25) | html | pdf |
  - 1.3.2. The mathematical theory of the Fourier transformation (pp. 25-49) | html | pdf |
    - 1.3.2.1. Introduction (pp. 25-26) | html | pdf |
    - 1.3.2.2. Preliminary notions and notation (pp. 26-28) | html | pdf |
      - 1.3.2.2.1. Metric and topological notions in $[{\bb R}^{n}]$ (p. 26) | html | pdf |
      - 1.3.2.2.2. Functions over $[{\bb R}^{n}]$ (pp. 26-27) | html | pdf |
      - 1.3.2.2.3. Multi-index notation (p. 27) | html | pdf |
      - 1.3.2.2.4. Integration, $[L^{p}]$ spaces (p. 27) | html | pdf |
      - 1.3.2.2.5. Tensor products. Fubini's theorem (pp. 27-28) | html | pdf |
      - 1.3.2.2.6. Topology in function spaces  (p. 28) | html | pdf |
        1.3.2.2.6.1. General topology  (p. 28) | html | pdf |
        1.3.2.2.6.2. Topological vector spaces  (p. 28) | html | pdf |
    - 1.3.2.3. Elements of the theory of distributions (pp. 28-34) | html | pdf |
      - 1.3.2.3.1. Origins (p. 29) | html | pdf |
      - 1.3.2.3.2. Rationale (p. 29) | html | pdf |
      - 1.3.2.3.3. Test-function spaces  (pp. 29-30) | html | pdf |
        1.3.2.3.3.1. Topology on $[{\scr E}(\Omega)]$   (pp. 29-30) | html | pdf |
        1.3.2.3.3.2. Topology on $[{\scr D}_{k} (\Omega)]$   (p. 30) | html | pdf |
        1.3.2.3.3.3. Topology on $[{\scr D}(\Omega)]$   (p. 30) | html | pdf |
        1.3.2.3.3.4. Topologies on $[{\scr E}^{(m)}, {\scr D}_{k}^{(m)},{\scr D}^{(m)}]$   (p. 30) | html | pdf |
      - 1.3.2.3.4. Definition of distributions (p. 30) | html | pdf |
      - 1.3.2.3.5. First examples of distributions (p. 30) | html | pdf |
      - 1.3.2.3.6. Distributions associated to locally integrable functions (pp. 30-31) | html | pdf |
      - 1.3.2.3.7. Support of a distribution (p. 31) | html | pdf |
      - 1.3.2.3.8. Convergence of distributions (p. 31) | html | pdf |
      - 1.3.2.3.9. Operations on distributions  (pp. 31-34) | html | pdf |
        1.3.2.3.9.1. Differentiation  (pp. 31-32) | html | pdf |
        1.3.2.3.9.2. Integration of distributions in dimension 1  (p. 32) | html | pdf |
        1.3.2.3.9.3. Multiplication of distributions by functions  (p. 32) | html | pdf |
        1.3.2.3.9.4. Division of distributions by functions  (p. 33) | html | pdf |
        1.3.2.3.9.5. Transformation of coordinates  (p. 33) | html | pdf |
        1.3.2.3.9.6. Tensor product of distributions  (p. 33) | html | pdf |
        1.3.2.3.9.7. Convolution of distributions  (pp. 33-34) | html | pdf |
    - 1.3.2.4. Fourier transforms of functions (pp. 34-38) | html | pdf |
      - 1.3.2.4.1. Introduction (p. 34) | html | pdf |
      - 1.3.2.4.2. Fourier transforms in $[L^{1}]$   (pp. 35-37) | html | pdf |
        1.3.2.4.2.1. Linearity  (p. 35) | html | pdf |
        1.3.2.4.2.2. Effect of affine coordinate transformations  (p. 35) | html | pdf |
        1.3.2.4.2.3. Conjugate symmetry  (p. 35) | html | pdf |
        1.3.2.4.2.4. Tensor product property  (p. 35) | html | pdf |
        1.3.2.4.2.5. Convolution property  (p. 35) | html | pdf |
        1.3.2.4.2.6. Reciprocity property  (p. 35) | html | pdf |
        1.3.2.4.2.7. Riemann–Lebesgue lemma  (p. 35) | html | pdf |
        1.3.2.4.2.8. Differentiation  (pp. 35-36) | html | pdf |
        1.3.2.4.2.9. Decrease at infinity  (p. 36) | html | pdf |
        1.3.2.4.2.10. The Paley–Wiener theorem  (p. 36) | html | pdf |
      - 1.3.2.4.3. Fourier transforms in $[L^{2}]$   (pp. 36-37) | html | pdf |
        1.3.2.4.3.1. Invariance of $[L^{2}]$   (p. 36) | html | pdf |
        1.3.2.4.3.2. Reciprocity  (p. 36) | html | pdf |
        1.3.2.4.3.3. Isometry  (p. 36) | html | pdf |
        1.3.2.4.3.4. Eigenspace decomposition of $[L^{2}]$   (p. 36) | html | pdf |
        1.3.2.4.3.5. The convolution theorem and the isometry property  (pp. 36-37) | html | pdf |
      - 1.3.2.4.4. Fourier transforms in $[{\scr S}]$   (pp. 37-38) | html | pdf |
        1.3.2.4.4.1. Definition and properties of $[{\scr S}]$   (p. 37) | html | pdf |
        1.3.2.4.4.2. Gaussian functions and Hermite functions  (pp. 37-38) | html | pdf |
        1.3.2.4.4.3. Heisenberg's inequality, Hardy's theorem  (p. 38) | html | pdf |
        1.3.2.4.4.4. Symmetry property  (p. 38) | html | pdf |
      - 1.3.2.4.5. Various writings of Fourier transforms (p. 38) | html | pdf |
      - 1.3.2.4.6. Tables of Fourier transforms (p. 38) | html | pdf |
    - 1.3.2.5. Fourier transforms of tempered distributions (pp. 38-40) | html | pdf |
      - 1.3.2.5.1. Introduction (pp. 38-39) | html | pdf |
      - 1.3.2.5.2. $[{\scr S}]$ as a test-function space (p. 39) | html | pdf |
      - 1.3.2.5.3. Definition and examples of tempered distributions (p. 39) | html | pdf |
      - 1.3.2.5.4. Fourier transforms of tempered distributions (p. 39) | html | pdf |
      - 1.3.2.5.5. Transposition of basic properties (p. 39) | html | pdf |
      - 1.3.2.5.6. Transforms of δ-functions (pp. 39-40) | html | pdf |
      - 1.3.2.5.7. Reciprocity theorem (p. 40) | html | pdf |
      - 1.3.2.5.8. Multiplication and convolution (p. 40) | html | pdf |
      - 1.3.2.5.9. $[L^{2}]$ aspects, Sobolev spaces (p. 40) | html | pdf |
    - 1.3.2.6. Periodic distributions and Fourier series (pp. 40-45) | html | pdf |
      - 1.3.2.6.1. Terminology (pp. 40-41) | html | pdf |
      - 1.3.2.6.2. $[{\bb Z}^{n}]$ -periodic distributions in $[{\bb R}^{n}]$ (p. 41) | html | pdf |
      - 1.3.2.6.3. Identification with distributions over $[{\bb R}^{n}/{\bb Z}^{n}]$ (p. 41) | html | pdf |
      - 1.3.2.6.4. Fourier transforms of periodic distributions (pp. 41-42) | html | pdf |
      - 1.3.2.6.5. The case of non-standard period lattices (p. 42) | html | pdf |
      - 1.3.2.6.6. Duality between periodization and sampling (p. 42) | html | pdf |
      - 1.3.2.6.7. The Poisson summation formula (pp. 42-43) | html | pdf |
      - 1.3.2.6.8. Convolution of Fourier series (p. 43) | html | pdf |
      - 1.3.2.6.9. Toeplitz forms, Szegö's theorem  (pp. 43-44) | html | pdf |
        1.3.2.6.9.1. Toeplitz forms  (p. 43) | html | pdf |
        1.3.2.6.9.2. The Toeplitz–Carathéodory–Herglotz theorem  (p. 43) | html | pdf |
        1.3.2.6.9.3. Asymptotic distribution of eigenvalues of Toeplitz forms  (pp. 43-44) | html | pdf |
        1.3.2.6.9.4. Consequences of Szegö's theorem  (p. 44) | html | pdf |
      - 1.3.2.6.10. Convergence of Fourier series  (pp. 44-45) | html | pdf |
        1.3.2.6.10.1. Classical $[L^{1}]$ theory  (pp. 44-45) | html | pdf |
        1.3.2.6.10.2. Classical $[L^{2}]$ theory  (p. 45) | html | pdf |
        1.3.2.6.10.3. The viewpoint of distribution theory  (p. 45) | html | pdf |
    - 1.3.2.7. The discrete Fourier transformation (pp. 45-49) | html | pdf |
      - 1.3.2.7.1. Shannon's sampling theorem and interpolation formula (pp. 45-46) | html | pdf |
      - 1.3.2.7.2. Duality between subdivision and decimation of period lattices  (pp. 46-47) | html | pdf |
        1.3.2.7.2.1. Geometric description of sublattices  (p. 46) | html | pdf |
        1.3.2.7.2.2. Sublattice relations for reciprocal lattices  (p. 46) | html | pdf |
        1.3.2.7.2.3. Relation between lattice distributions  (pp. 46-47) | html | pdf |
        1.3.2.7.2.4. Relation between Fourier transforms  (p. 47) | html | pdf |
        1.3.2.7.2.5. Sublattice relations in terms of periodic distributions  (p. 47) | html | pdf |
      - 1.3.2.7.3. Discretization of the Fourier transformation (pp. 47-48) | html | pdf |
      - 1.3.2.7.4. Matrix representation of the discrete Fourier transform (DFT) (p. 49) | html | pdf |
      - 1.3.2.7.5. Properties of the discrete Fourier transform (p. 49) | html | pdf |
  - 1.3.3. Numerical computation of the discrete Fourier transform (pp. 49-58) | html | pdf |
    - 1.3.3.1. Introduction (pp. 49-50) | html | pdf |
    - 1.3.3.2. One-dimensional algorithms (pp. 50-55) | html | pdf |
      - 1.3.3.2.1. The Cooley–Tukey algorithm (pp. 50-51) | html | pdf |
      - 1.3.3.2.2. The Good (or prime factor) algorithm  (pp. 51-52) | html | pdf |
        1.3.3.2.2.1. Ring structure on $[{\bb Z}/N{\bb Z}]$   (p. 51) | html | pdf |
        1.3.3.2.2.2. The Chinese remainder theorem  (pp. 51-52) | html | pdf |
        1.3.3.2.2.3. The prime factor algorithm  (p. 52) | html | pdf |
      - 1.3.3.2.3. The Rader algorithm  (pp. 53-54) | html | pdf |
        1.3.3.2.3.1. N an odd prime  (p. 53) | html | pdf |
        1.3.3.2.3.2. N a power of an odd prime  (p. 53) | html | pdf |
        1.3.3.2.3.3. N a power of 2  (pp. 53-54) | html | pdf |
      - 1.3.3.2.4. The Winograd algorithms (pp. 54-55) | html | pdf |
    - 1.3.3.3. Multidimensional algorithms (pp. 55-58) | html | pdf |
      - 1.3.3.3.1. The method of successive one-dimensional transforms (p. 55) | html | pdf |
      - 1.3.3.3.2. Multidimensional factorization  (pp. 55-57) | html | pdf |
        1.3.3.3.2.1. Multidimensional Cooley–Tukey factorization  (pp. 55-56) | html | pdf |
        1.3.3.3.2.2. Multidimensional prime factor algorithm  (p. 56) | html | pdf |
        1.3.3.3.2.3. Nesting of Winograd small FFTs  (pp. 56-57) | html | pdf |
        1.3.3.3.2.4. The Nussbaumer–Quandalle algorithm  (p. 57) | html | pdf |
      - 1.3.3.3.3. Global algorithm design  (pp. 57-58) | html | pdf |
        1.3.3.3.3.1. From local pieces to global algorithms  (pp. 57-58) | html | pdf |
        1.3.3.3.3.2. Computer architecture considerations  (p. 58) | html | pdf |
        1.3.3.3.3.3. The Johnson–Burrus family of algorithms  (p. 58) | html | pdf |
  - 1.3.4. Crystallographic applications of Fourier transforms (pp. 58-98) | html | pdf |
    - 1.3.4.1. Introduction (pp. 58-59) | html | pdf |
    - 1.3.4.2. Crystallographic Fourier transform theory (pp. 59-71) | html | pdf |
      - 1.3.4.2.1. Crystal periodicity  (pp. 59-64) | html | pdf |
        1.3.4.2.1.1. Period lattice, reciprocal lattice and structure factors  (pp. 59-60) | html | pdf |
        1.3.4.2.1.2. Structure factors in terms of form factors  (p. 60) | html | pdf |
        1.3.4.2.1.3. Fourier series for the electron density and its summation  (p. 60) | html | pdf |
        1.3.4.2.1.4. Friedel's law, anomalous scatterers  (p. 60) | html | pdf |
        1.3.4.2.1.5. Parseval's identity and other $[L^{2}]$ theorems  (p. 61) | html | pdf |
        1.3.4.2.1.6. Convolution, correlation and Patterson function  (p. 61) | html | pdf |
        1.3.4.2.1.7. Sampling theorems, continuous transforms, interpolation  (pp. 61-62) | html | pdf |
        1.3.4.2.1.8. Sections and projections  (pp. 62-63) | html | pdf |
        1.3.4.2.1.9. Differential syntheses  (p. 63) | html | pdf |
        1.3.4.2.1.10. Toeplitz forms, determinantal inequalities and Szegö's theorem  (pp. 63-64) | html | pdf |
      - 1.3.4.2.2. Crystal symmetry  (pp. 64-71) | html | pdf |
        1.3.4.2.2.1. Crystallographic groups  (p. 64) | html | pdf |
        1.3.4.2.2.2. Groups and group actions  (pp. 64-66) | html | pdf |
        1.3.4.2.2.3. Classification of crystallographic groups  (pp. 66-67) | html | pdf |
        1.3.4.2.2.4. Crystallographic group action in real space  (pp. 67-68) | html | pdf |
        1.3.4.2.2.5. Crystallographic group action in reciprocal space  (p. 68) | html | pdf |
        1.3.4.2.2.6. Structure-factor calculation  (pp. 68-69) | html | pdf |
        1.3.4.2.2.7. Electron-density calculations  (p. 69) | html | pdf |
        1.3.4.2.2.8. Parseval's theorem with crystallographic symmetry  (p. 69) | html | pdf |
        1.3.4.2.2.9. Convolution theorems with crystallographic symmetry  (p. 70) | html | pdf |
        1.3.4.2.2.10. Correlation and Patterson functions  (pp. 70-71) | html | pdf |
    - 1.3.4.3. Crystallographic discrete Fourier transform algorithms (pp. 71-84) | html | pdf |
      - 1.3.4.3.1. Historical introduction (pp. 71-72) | html | pdf |
      - 1.3.4.3.2. Defining relations and symmetry considerations (pp. 72-73) | html | pdf |
      - 1.3.4.3.3. Interaction between symmetry and decomposition (p. 73) | html | pdf |
      - 1.3.4.3.4. Interaction between symmetry and factorization  (pp. 73-79) | html | pdf |
        1.3.4.3.4.1. Multidimensional Cooley–Tukey factorization  (pp. 74-76) | html | pdf |
        1.3.4.3.4.2. Multidimensional Good factorization  (p. 76) | html | pdf |
        1.3.4.3.4.3. Crystallographic extension of the Rader/Winograd factorization  (pp. 76-79) | html | pdf |
      - 1.3.4.3.5. Treatment of conjugate and parity-related symmetry properties  (pp. 79-82) | html | pdf |
        1.3.4.3.5.1. Hermitian-symmetric or real-valued transforms  (pp. 79-80) | html | pdf |
        1.3.4.3.5.2. Hermitian-antisymmetric or pure imaginary transforms  (p. 80) | html | pdf |
        1.3.4.3.5.3. Complex symmetric and antisymmetric transforms  (pp. 80-81) | html | pdf |
        1.3.4.3.5.4. Real symmetric transforms  (p. 81) | html | pdf |
        1.3.4.3.5.5. Real antisymmetric transforms  (p. 82) | html | pdf |
        1.3.4.3.5.6. Generalized multiplexing  (p. 82) | html | pdf |
      - 1.3.4.3.6. Global crystallographic algorithms  (pp. 82-84) | html | pdf |
        1.3.4.3.6.1. Triclinic groups  (p. 82) | html | pdf |
        1.3.4.3.6.2. Monoclinic groups  (p. 82) | html | pdf |
        1.3.4.3.6.3. Orthorhombic groups  (pp. 82-83) | html | pdf |
        1.3.4.3.6.4. Trigonal, tetragonal and hexagonal groups  (p. 83) | html | pdf |
        1.3.4.3.6.5. Cubic groups  (p. 83) | html | pdf |
        1.3.4.3.6.6. Treatment of centred lattices  (p. 83) | html | pdf |
        1.3.4.3.6.7. Programming considerations  (pp. 83-84) | html | pdf |
    - 1.3.4.4. Basic crystallographic computations (pp. 84-93) | html | pdf |
      - 1.3.4.4.1. Introduction (p. 84) | html | pdf |
      - 1.3.4.4.2. Fourier synthesis of electron-density maps (p. 84) | html | pdf |
      - 1.3.4.4.3. Fourier analysis of modified electron-density maps  (pp. 84-86) | html | pdf |
        1.3.4.4.3.1. Squaring  (p. 84) | html | pdf |
        1.3.4.4.3.2. Other non-linear operations  (p. 84) | html | pdf |
        1.3.4.4.3.3. Solvent flattening  (pp. 84-85) | html | pdf |
        1.3.4.4.3.4. Molecular averaging by noncrystallographic symmetries  (pp. 85-86) | html | pdf |
        1.3.4.4.3.5. Molecular-envelope transforms via Green's theorem  (p. 86) | html | pdf |
      - 1.3.4.4.4. Structure factors from model atomic parameters (p. 86) | html | pdf |
      - 1.3.4.4.5. Structure factors via model electron-density maps (pp. 86-87) | html | pdf |
      - 1.3.4.4.6. Derivatives for variational phasing techniques (pp. 87-88) | html | pdf |
      - 1.3.4.4.7. Derivatives for model refinement  (pp. 88-92) | html | pdf |
        1.3.4.4.7.1. The method of least squares  (p. 88) | html | pdf |
        1.3.4.4.7.2. Booth's differential Fourier syntheses  (pp. 88-89) | html | pdf |
        1.3.4.4.7.3. Booth's method of steepest descents  (p. 89) | html | pdf |
        1.3.4.4.7.4. Cochran's Fourier method  (pp. 89-90) | html | pdf |
        1.3.4.4.7.5. Cruickshank's modified Fourier method  (p. 90) | html | pdf |
        1.3.4.4.7.6. Agarwal's FFT implementation of the Fourier method  (pp. 90-91) | html | pdf |
        1.3.4.4.7.7. Lifchitz's reformulation  (p. 91) | html | pdf |
        1.3.4.4.7.8. A simplified derivation  (p. 91) | html | pdf |
        1.3.4.4.7.9. Discussion of macromolecular refinement techniques  (p. 92) | html | pdf |
        1.3.4.4.7.10. Sampling considerations  (p. 92) | html | pdf |
      - 1.3.4.4.8. Miscellaneous correlation functions (p. 92) | html | pdf |
    - 1.3.4.5. Related applications (pp. 93-98) | html | pdf |
      - 1.3.4.5.1. Helical diffraction  (pp. 93-94) | html | pdf |
        1.3.4.5.1.1. Circular harmonic expansions in polar coordinates  (p. 93) | html | pdf |
        1.3.4.5.1.2. The Fourier transform in polar coordinates  (p. 93) | html | pdf |
        1.3.4.5.1.3. The transform of an axially periodic fibre  (p. 93) | html | pdf |
        1.3.4.5.1.4. Helical symmetry and associated selection rules  (pp. 93-94) | html | pdf |
      - 1.3.4.5.2. Application to probability theory and direct methods  (pp. 94-98) | html | pdf |
        1.3.4.5.2.1. Analytical methods of probability theory  (pp. 94-96) | html | pdf |
        1.3.4.5.2.2. The statistical theory of phase determination  (pp. 96-98) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 1.3.3.1. A few global algorithms for computing a 400-point DFT (p. 58) | html | pdf |
- 1.4. Symmetry in reciprocal space (pp. 99-161) | html | pdf | chapter contents |
  U. Shmueli, S. R. Hall and R. W. Grosse-Kunstleve
  - 1.4.1. Introduction (p. 99) | html | pdf |
    U. Shmueli
  - 1.4.2. Effects of symmetry on the Fourier image of the crystal (pp. 99-102) | html | pdf |
    U. Shmueli
    - 1.4.2.1. Point-group symmetry of the reciprocal lattice (p. 99) | html | pdf |
    - 1.4.2.2. Relationship between structure factors at symmetry-related points of the reciprocal lattice (pp. 99-101) | html | pdf |
    - 1.4.2.3. Symmetry factors for space-group-specific Fourier summations (p. 101) | html | pdf |
    - 1.4.2.4. Symmetry factors for space-group-specific structure-factor formulae (pp. 101-102) | html | pdf |
  - 1.4.3. Structure-factor tables (pp. 102-104) | html | pdf |
    U. Shmueli
    - 1.4.3.1. Some general remarks (p. 102) | html | pdf |
    - 1.4.3.2. Preparation of the structure-factor tables (p. 102) | html | pdf |
    - 1.4.3.3. Symbolic representation of A and B (pp. 102-103) | html | pdf |
    - 1.4.3.4. Arrangement of the tables (pp. 103-104) | html | pdf |
  - 1.4.4. Symmetry in reciprocal space: space-group tables (pp. 104-106) | html | pdf |
    U. Shmueli
    - 1.4.4.1. Introduction (p. 104) | html | pdf |
    - 1.4.4.2. Arrangement of the space-group tables (p. 104) | html | pdf |
    - 1.4.4.3. Effect of direct-space transformations (pp. 104-105) | html | pdf |
    - 1.4.4.4. Symmetry in Fourier space (p. 105) | html | pdf |
    - 1.4.4.5. Relationships between direct and reciprocal Bravais lattices (pp. 105-106) | html | pdf |
  - Appendix 1.4.1. Comments on the preparation and usage of the tables (pp. 106-107) | html | pdf |
    U. Shmueli
  - Appendix 1.4.2. Space-group symbols for numeric and symbolic computations (pp. 107-119) | html | pdf |
    - A1.4.2.1. Introduction (p. 107) | html | pdf |
      U. Shmueli, S. R. Hall and R. W. Grosse-Kunstleve
    - A1.4.2.2. Explicit symbols (pp. 108-112) | html | pdf |
      U. Shmueli
    - A1.4.2.3. Hall symbols (pp. 112-114) | html | pdf |
      S. R. Hall and R. W. Grosse-Kunstleve
      - A1.4.2.3.1. Default axes (p. 114) | html | pdf |
      - A1.4.2.3.2. Example matrices (p. 114) | html | pdf |
  - Appendix 1.4.3. Structure-factor tables (pp. 120-149) | html | pdf |
    U. Shmueli
  - Appendix 1.4.4. Crystallographic space groups in reciprocal space (pp. 150-161) | html | pdf |
    U. Shmueli
  - References | html | pdf |
  - Tables
    - Table 1.4.4.1. Correspondence between types of centring in direct and reciprocal lattices (p. 106) | html | pdf |
    - Table A1.4.2.1. Explicit symbols (pp. 109-111) | html | pdf |
    - Table A1.4.2.2. Lattice symbol L (p. 112) | html | pdf |
    - Table A1.4.2.3. Translation symbol T (p. 112) | html | pdf |
    - Table A1.4.2.4. Rotation matrices for principal axes (p. 113) | html | pdf |
    - Table A1.4.2.5. Rotation matrices for face-diagonal axes (p. 113) | html | pdf |
    - Table A1.4.2.6. Rotation matrix for the body-diagonal axis (p. 113) | html | pdf |
    - Table A1.4.2.7. Hall symbols (pp. 115-119) | html | pdf |
    - Table A1.4.3.1. Plane groups (p. 120) | html | pdf |
    - Table A1.4.3.2. Triclinic space groups (p. 120) | html | pdf |
    - Table A1.4.3.3. Monoclinic space groups (pp. 121-122) | html | pdf |
    - Table A1.4.3.4. Orthorhombic space groups (pp. 123-125) | html | pdf |
    - Table A1.4.3.5. Tetragonal space groups (pp. 126-136) | html | pdf |
    - Table A1.4.3.6. Trigonal and hexagonal space groups (pp. 137-143) | html | pdf |
    - Table A1.4.3.7. Cubic space groups (pp. 144-149) | html | pdf |
    - Table A1.4.4.1. Crystallographic space groups in reciprocal space (pp. 150-161) | html | pdf |
- 1.5. Crystallographic viewpoints in the classification of space-group representations (pp. 162-188) | html | pdf | chapter contents |
  M. I. Aroyo and H. Wondratschek
  - 1.5.1. List of symbols (p. 162) | html | pdf |
  - 1.5.2. Introduction (p. 162) | html | pdf |
  - 1.5.3. Basic concepts (pp. 162-165) | html | pdf |
    - 1.5.3.1. Representations of finite groups (pp. 162-163) | html | pdf |
    - 1.5.3.2. Space groups (pp. 163-164) | html | pdf |
    - 1.5.3.3. Representations of the translation group $[{\cal T}]$ and the reciprocal lattice (pp. 164-165) | html | pdf |
    - 1.5.3.4. Irreducible representations of space groups and the reciprocal-space group (p. 165) | html | pdf |
  - 1.5.4. Conventions in the classification of space-group irreps (pp. 165-168) | html | pdf |
    - 1.5.4.1. Fundamental regions (pp. 165-166) | html | pdf |
    - 1.5.4.2. Minimal domains (pp. 166-167) | html | pdf |
    - 1.5.4.3. Wintgen positions (pp. 167-168) | html | pdf |
  - 1.5.5. Examples and conclusions (pp. 168-176) | html | pdf |
    - 1.5.5.1. Examples (p. 168) | html | pdf |
    - 1.5.5.2. Results (pp. 169-171) | html | pdf |
    - 1.5.5.3. Parameter ranges (pp. 171-172) | html | pdf |
    - 1.5.5.4. Conclusions (pp. 172-176) | html | pdf |
  - Appendix 1.5.1. Reciprocal-space groups $[{\cal G}^{*}]$ (p. 176) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 1.5.5.1. Symmorphic space group $[Fm\bar{3}m]$ (p. 169) | html | pdf |
    - Fig. 1.5.5.2. Symmorphic space group $[F\bar{3}m]$ (p. 171) | html | pdf |
    - Fig. 1.5.5.3. (a), (b) Symmorphic space group (p. 173) | html | pdf |
    - Fig. 1.5.5.4. Symmorphic space group Imm2 (p. 177) | html | pdf |
  - Tables
    - Table 1.5.4.1. Conventional coefficients $[(k_{i})^{T}]$ of k expressed by the adjusted coefficients $[(k_{ai})]$ of IT A for the different Bravais types of lattices in direct space (p. 167) | html | pdf |
    - Table 1.5.4.2. Primitive coefficients $[(k_{pi})^{T}]$ of k from CDML expressed by the adjusted coefficients $[(k_{ai})]$ of IT A for the different Bravais types of lattices in direct space (p. 167) | html | pdf |
    - Table 1.5.5.1. The k-vector types for the space groups $[Im\bar{3} m]$ and $[Ia\bar{3} d]$ (p. 168) | html | pdf |
    - Table 1.5.5.2. The k-vector types for the space groups $[Im\bar{3}]$ and $[Ia\bar{3}]$ (p. 170) | html | pdf |
    - Table 1.5.5.3. The k-vector types for the space groups , , $[I4_{1}/amd]$ and $[I4_{1}/acd]$ (p. 172) | html | pdf |
    - Table 1.5.5.4. The k-vector types for the space groups Fmm2 and Fdd2 (pp. 174-175) | html | pdf |

Reciprocal space in crystal-structure determination
- 2.1. Statistical properties of the weighted reciprocal lattice (pp. 190-209) | html | pdf | chapter contents |
  U. Shmueli and A. J. C. Wilson
  - 2.1.1. Introduction (p. 190) | html | pdf |
  - 2.1.2. The average intensity of general reflections (pp. 190-191) | html | pdf |
    - 2.1.2.1. Mathematical background (pp. 190-191) | html | pdf |
    - 2.1.2.2. Physical background (p. 191) | html | pdf |
    - 2.1.2.3. An approximation for organic compounds (p. 191) | html | pdf |
    - 2.1.2.4. Effect of centring (p. 191) | html | pdf |
  - 2.1.3. The average intensity of zones and rows (pp. 191-192) | html | pdf |
    - 2.1.3.1. Symmetry elements producing systematic absences (pp. 191-192) | html | pdf |
    - 2.1.3.2. Symmetry elements not producing systematic absences (p. 192) | html | pdf |
    - 2.1.3.3. More than one symmetry element (p. 192) | html | pdf |
  - 2.1.4. Probability density distributions – mathematical preliminaries (pp. 192-195) | html | pdf |
    - 2.1.4.1. Characteristic functions (pp. 192-193) | html | pdf |
    - 2.1.4.2. The cumulant-generating function (pp. 193-194) | html | pdf |
    - 2.1.4.3. The central-limit theorem (pp. 194-195) | html | pdf |
    - 2.1.4.4. Conditions of validity (p. 195) | html | pdf |
    - 2.1.4.5. Non-independent variables (p. 195) | html | pdf |
  - 2.1.5. Ideal probability density distributions (pp. 195-197) | html | pdf |
    - 2.1.5.1. Ideal acentric distributions (pp. 195-196) | html | pdf |
    - 2.1.5.2. Ideal centric distributions (p. 196) | html | pdf |
    - 2.1.5.3. Effect of other symmetry elements on the ideal acentric and centric distributions (p. 196) | html | pdf |
    - 2.1.5.4. Other ideal distributions (p. 196) | html | pdf |
    - 2.1.5.5. Relation to distributions of I (p. 196) | html | pdf |
    - 2.1.5.6. Cumulative distribution functions (pp. 196-197) | html | pdf |
  - 2.1.6. Distributions of sums, averages and ratios (pp. 197-199) | html | pdf |
    - 2.1.6.1. Distributions of sums and averages (p. 197) | html | pdf |
    - 2.1.6.2. Distribution of ratios (pp. 197-198) | html | pdf |
    - 2.1.6.3. Intensities scaled to the local average (p. 198) | html | pdf |
    - 2.1.6.4. The use of normal approximations (pp. 198-199) | html | pdf |
  - 2.1.7. Non-ideal distributions: the correction-factor approach (pp. 199-203) | html | pdf |
    - 2.1.7.1. Introduction (p. 199) | html | pdf |
    - 2.1.7.2. Mathematical background (pp. 199-200) | html | pdf |
    - 2.1.7.3. Application to centric and acentric distributions (pp. 200-203) | html | pdf |
    - 2.1.7.4. Fourier versus Hermite approximations (p. 203) | html | pdf |
  - 2.1.8. Non-ideal distributions: the Fourier method (pp. 203-209) | html | pdf |
    - 2.1.8.1. General representations of p.d.f.'s of by Fourier series (pp. 203-204) | html | pdf |
    - 2.1.8.2. Fourier–Bessel series (pp. 204-205) | html | pdf |
    - 2.1.8.3. Simple examples (p. 205) | html | pdf |
    - 2.1.8.4. A more complicated example (pp. 205-206) | html | pdf |
    - 2.1.8.5. Atomic characteristic functions (pp. 206-208) | html | pdf |
    - 2.1.8.6. Other non-ideal Fourier p.d.f.'s (p. 208) | html | pdf |
    - 2.1.8.7. Comparison of the correction-factor and Fourier approaches (pp. 208-209) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 2.1.7.1. Atomic heterogeneity and intensity statistics (p. 199) | html | pdf |
  - Tables
    - Table 2.1.3.1. Intensity-distribution effects of symmetry elements causing systematic absences (p. 191) | html | pdf |
    - Table 2.1.3.2. Intensity-distribution effects of symmetry elements not causing systematic absences (p. 192) | html | pdf |
    - Table 2.1.3.3. Average multiples for the 32 point groups (modified from Rogers, 1950). (p. 193) | html | pdf |
    - Table 2.1.5.1. Some properties of gamma and beta distributions (p. 197) | html | pdf |
    - Table 2.1.7.1. Some even absolute moments of the trigonometric structure factor (pp. 201-202) | html | pdf |
    - Table 2.1.7.2. Closed expressions for $[\gamma_{2k}]$ [equation (2.1.7.11)] for space groups of low symmetry (p. 203) | html | pdf |
    - Table 2.1.8.1. Atomic contributions to characteristic functions for (pp. 207-208) | html | pdf |
- 2.2. Direct methods (pp. 210-234) | html | pdf | chapter contents |
  C. Giacovazzo
  - 2.2.1. List of symbols and abbreviations (p. 210) | html | pdf |
  - 2.2.2. Introduction (p. 210) | html | pdf |
  - 2.2.3. Origin specification (pp. 210-215) | html | pdf |
  - 2.2.4. Normalized structure factors (pp. 215-217) | html | pdf |
    - 2.2.4.1. Definition of normalized structure factor (pp. 215-216) | html | pdf |
    - 2.2.4.2. Definition of quasi-normalized structure factor (p. 216) | html | pdf |
    - 2.2.4.3. The calculation of normalized structure factors (pp. 216-217) | html | pdf |
    - 2.2.4.4. Probability distributions of normalized structure factors (p. 217) | html | pdf |
  - 2.2.5. Phase-determining formulae (pp. 217-225) | html | pdf |
    - 2.2.5.1. Inequalities among structure factors (pp. 217-218) | html | pdf |
    - 2.2.5.2. Probabilistic phase relationships for structure invariants (p. 218) | html | pdf |
    - 2.2.5.3. Triplet relationships (pp. 218-219) | html | pdf |
    - 2.2.5.4. Triplet relationships using structural information (p. 220) | html | pdf |
    - 2.2.5.5. Quartet phase relationships (pp. 220-222) | html | pdf |
    - 2.2.5.6. Quintet phase relationships (pp. 222-223) | html | pdf |
    - 2.2.5.7. Determinantal formulae (pp. 223-224) | html | pdf |
    - 2.2.5.8. Algebraic relationships for structure seminvariants (p. 224) | html | pdf |
    - 2.2.5.9. Formulae estimating one-phase structure seminvariants of the first rank (pp. 224-225) | html | pdf |
    - 2.2.5.10. Formulae estimating two-phase structure seminvariants of the first rank (p. 225) | html | pdf |
  - 2.2.6. Direct methods in real and reciprocal space: Sayre's equation (pp. 225-227) | html | pdf |
  - 2.2.7. Scheme of procedure for phase determination (pp. 227-228) | html | pdf |
  - 2.2.8. Other multisolution methods applied to small molecules (pp. 228-230) | html | pdf |
  - 2.2.9. Some references to direct-methods packages (p. 230) | html | pdf |
  - 2.2.10. Direct methods in macromolecular crystallography (pp. 231-234) | html | pdf |
    - 2.2.10.1. Introduction (p. 231) | html | pdf |
    - 2.2.10.2. Ab initio direct phasing of proteins (pp. 231-232) | html | pdf |
    - 2.2.10.3. Integration of direct methods with isomorphous replacement techniques (p. 232) | html | pdf |
    - 2.2.10.4. Integration of anomalous-dispersion techniques with direct methods (pp. 232-234) | html | pdf |
      - 2.2.10.4.1. One-wavelength techniques (p. 233) | html | pdf |
      - 2.2.10.4.2. The SIRAS, MIRAS and MAD cases (pp. 233-234) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 2.2.4.1. Probability density functions for cs. and ncs. crystals (p. 216) | html | pdf |
    - Fig. 2.2.4.2. Cumulative distribution functions for cs. and ncs. crystals (p. 216) | html | pdf |
    - Fig. 2.2.5.1. Curves of (2.2.5.6) for some values of $[2\sigma_{3} \sigma_{2}^{-3/2} |E_{\bf h} E_{\bf k} E_{{\bf h} - {\bf k}}|]$ (p. 219) | html | pdf |
    - Fig. 2.2.5.2. Variance (in square radians) as a function of α (p. 219) | html | pdf |
    - Fig. 2.2.5.3. Distributions (2.2.5.18) (solid curve) and (2.2.5.20) (dashed curve) for the indicated values in three typical cases (p. 221) | html | pdf |
    - Fig. 2.2.7.1. The form of w as given by (2.2.7.2) (p. 227) | html | pdf |
  - Tables
    - Table 2.2.3.1. Allowed origin translations, seminvariant moduli and phases for centrosymmetric primitive space groups (p. 211) | html | pdf |
    - Table 2.2.3.2. Allowed origin translations, seminvariant moduli and phases for noncentrosymmetric primitive space groups (pp. 212-213) | html | pdf |
    - Table 2.2.3.3. Allowed origin translations, seminvariant moduli and phases for centrosymmetric non-primitive space groups (p. 214) | html | pdf |
    - Table 2.2.3.4. Allowed origin translations, seminvariant moduli and phases for noncentrosymmetric non-primitive space groups (pp. 214-215) | html | pdf |
    - Table 2.2.4.1. Moments of the distributions (2.2.4.4) and (2.2.4.5) (p. 217) | html | pdf |
    - Table 2.2.5.1. List of quartets symmetry equivalent to $[\Phi = \Phi_{1}]$ in the class mmm (p. 222) | html | pdf |
    - Table 2.2.8.1. Magic-integer sequences for small numbers of phases (n) together with the number of sets produced and the root-mean-square error in the phases (p. 229) | html | pdf |
- 2.3. Patterson and molecular-replacement techniques (pp. 235-263) | html | pdf | chapter contents |
  M. G. Rossmann and E. Arnold
  - 2.3.1. Introduction (pp. 235-238) | html | pdf |
    - 2.3.1.1. Background (p. 235) | html | pdf |
    - 2.3.1.2. Limits to the number of resolved vectors (pp. 235-236) | html | pdf |
    - 2.3.1.3. Modifications: origin removal, sharpening etc. (pp. 236-237) | html | pdf |
    - 2.3.1.4. Homometric structures and the uniqueness of structure solutions; enantiomorphic solutions (p. 237) | html | pdf |
    - 2.3.1.5. The Patterson synthesis of the second kind (p. 238) | html | pdf |
  - 2.3.2. Interpretation of Patterson maps (pp. 238-242) | html | pdf |
    - 2.3.2.1. Simple solutions in the triclinic cell. Selection of the origin (pp. 238-239) | html | pdf |
    - 2.3.2.2. Harker sections (p. 239) | html | pdf |
    - 2.3.2.3. Finding heavy atoms (pp. 239-240) | html | pdf |
    - 2.3.2.4. Superposition methods. Image detection (pp. 240-241) | html | pdf |
    - 2.3.2.5. Systematic computerized Patterson vector-search procedures. Looking for rigid bodies (pp. 241-242) | html | pdf |
  - 2.3.3. Isomorphous replacement difference Pattersons (pp. 242-246) | html | pdf |
    - 2.3.3.1. Introduction (p. 242) | html | pdf |
    - 2.3.3.2. Finding heavy atoms with centrosymmetric projections (pp. 242-243) | html | pdf |
    - 2.3.3.3. Finding heavy atoms with three-dimensional methods (p. 243) | html | pdf |
    - 2.3.3.4. Correlation functions (pp. 243-244) | html | pdf |
    - 2.3.3.5. Interpretation of isomorphous difference Pattersons (pp. 244-245) | html | pdf |
    - 2.3.3.6. Direct structure determination from difference Pattersons (p. 245) | html | pdf |
    - 2.3.3.7. Isomorphism and size of the heavy-atom substitution (pp. 245-246) | html | pdf |
  - 2.3.4. Anomalous dispersion (pp. 246-248) | html | pdf |
    - 2.3.4.1. Introduction (p. 246) | html | pdf |
    - 2.3.4.2. The $[P_s({\bf u})]$ function (pp. 246-247) | html | pdf |
    - 2.3.4.3. The position of anomalous scatterers (pp. 247-248) | html | pdf |
  - 2.3.5. Noncrystallographic symmetry (pp. 248-250) | html | pdf |
    - 2.3.5.1. Definitions (pp. 248-249) | html | pdf |
    - 2.3.5.2. Interpretation of Pattersons in the presence of noncrystallographic symmetry (pp. 249-250) | html | pdf |
  - 2.3.6. Rotation functions (pp. 250-258) | html | pdf |
    - 2.3.6.1. Introduction (pp. 250-252) | html | pdf |
    - 2.3.6.2. Matrix algebra (pp. 252-253) | html | pdf |
    - 2.3.6.3. Symmetry (pp. 253-254) | html | pdf |
    - 2.3.6.4. Sampling, background and interpretation (pp. 254-255) | html | pdf |
    - 2.3.6.5. The fast rotation function (pp. 255-258) | html | pdf |
  - 2.3.7. Translation functions (pp. 258-260) | html | pdf |
    - 2.3.7.1. Introduction (pp. 258-259) | html | pdf |
    - 2.3.7.2. Position of a noncrystallographic element relating two unknown structures (p. 259) | html | pdf |
    - 2.3.7.3. Position of a known molecular structure in an unknown unit cell (pp. 259-260) | html | pdf |
    - 2.3.7.4. Position of a noncrystallographic symmetry element in a poorly defined electron-density map (p. 260) | html | pdf |
  - 2.3.8. Molecular replacement (pp. 260-262) | html | pdf |
    - 2.3.8.1. Using a known molecular fragment (pp. 260-261) | html | pdf |
    - 2.3.8.2. Using noncrystallographic symmetry for phase improvement (pp. 261-262) | html | pdf |
    - 2.3.8.3. Equivalence of real- and reciprocal-space molecular replacement (p. 262) | html | pdf |
  - 2.3.9. Conclusions (pp. 262-263) | html | pdf |
    - 2.3.9.1. Update (pp. 262-263) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 2.3.1.1. Effect of `sharpening' Patterson coefficients (p. 237) | html | pdf |
    - Fig. 2.3.1.2. (c) The point Patterson of the two homometric structures in (a) and (b) (p. 237) | html | pdf |
    - Fig. 2.3.2.1. Origin selection in the interpretation of a Patterson of a one-dimensional centrosymmetric structure (p. 238) | html | pdf |
    - Fig. 2.3.2.2. The c-axis projection of cuprous chloride azomethane complex (C₂H₆Cl₂Cu₂N₂) (p. 238) | html | pdf |
    - Fig. 2.3.2.3. Atoms ABCD, arranged as a quadrilateral, generate a Patterson which is the sum of the images of the quadrilateral when each atom is placed on the origin in turn (p. 241) | html | pdf |
    - Fig. 2.3.3.1. Three different cases which can occur in the relation of the native, $[{\bf F}_{N}]$ , and heavy-atom derivative, $[{\bf F}_{NH}]$ , structure factors for centrosymmetric reflections (p. 242) | html | pdf |
    - Fig. 2.3.3.2. Vector triangle showing the relationship between $[{\bf F}_{N}]$ , $[{\bf F}_{NH}]$ and $[{\bf F}_{H}]$ , where $[{\bf F}_{NH} = {\bf F}_{N} + {\bf F}_{H}]$ (p. 243) | html | pdf |
    - Fig. 2.3.3.3. A Patterson with coefficients $[({\bf F}_{NH1} - {\bf F}_{NH2})^{2}]$ will be equivalent to a Patterson whose coefficients are $[(AB)^{2}]$ (p. 243) | html | pdf |
    - Fig. 2.3.3.4. The phase α of the native compound (structure factor $[{\bf F}_{N}]$ ) is determined either as being equal to, or 180° out of phase with, the presumed heavy-atom contribution when only a single isomorphous compound is available (p. 244) | html | pdf |
    - Fig. 2.3.3.5. Let (a) be the original structure which contains three heavy atoms ABC in a noncentrosymmetric configuration (p. 245) | html | pdf |
    - Fig. 2.3.3.6. A plot of mean isomorphous differences as a function of resolution (p. 246) | html | pdf |
    - Fig. 2.3.4.1. (a) A model structure with an anomalous scatterer at A (p. 247) | html | pdf |
    - Fig. 2.3.4.2. Anomalous-dispersion effect for a molecule (p. 247) | html | pdf |
    - Fig. 2.3.5.1. The two-dimensional periodic design shows crystallographic twofold axes perpendicular to the page and local noncrystallographic rotation axes in the plane of the paper (p. 248) | html | pdf |
    - Fig. 2.3.5.2. The objects $[A_{1}B_{1}]$ and $[A_{2}B_{2}]$ are related by an improper rotation θ (p. 249) | html | pdf |
    - Fig. 2.3.5.3. The position of the twofold rotation axis which relates the two piglets is completely arbitrary (p. 249) | html | pdf |
    - Fig. 2.3.6.1. Shape of the interference function G for a spherical envelope of radius R at a distance H from the reciprocal-space origin (p. 252) | html | pdf |
    - Fig. 2.3.6.2. Relationships of the orthogonal axes $[X_{1}, X_{2}, X_{3}]$ to the crystallographic axes $[a_{1}, a_{2}, a_{3}]$ (p. 252) | html | pdf |
    - Fig. 2.3.6.3. Eulerian angles $[\theta_{1}, \theta_{2}, \theta_{3}]$ relating the rotated axes $[X'_{1}, X'_{2}, X'_{3}]$ to the original unrotated orthogonal axes $[X_{1}, X_{2}, X_{3}]$ (p. 253) | html | pdf |
    - Fig. 2.3.6.4. Variables ψ and φ are polar coordinates which specify a direction about which the axes may be rotated through an angle κ (p. 253) | html | pdf |
    - Fig. 2.3.6.5. Rotation space group diagram for the rotation function of a Pmmm Patterson function $[(P_{1})]$ against a Patterson function $[(P_{2})]$ (p. 254) | html | pdf |
    - Fig. 2.3.6.6. The locked rotation function, L, applied to the determination of the orientation of the common cold virus (p. 255) | html | pdf |
    - Fig. 2.3.7.1. Crosses represent atoms in a two-dimensional model structure (p. 259) | html | pdf |
    - Fig. 2.3.7.2. Vectors arising from the structure in Fig. 2.3.7.1 (p. 259) | html | pdf |
  - Tables
    - Table 2.3.1.1. Matrix representation of Patterson peaks (p. 236) | html | pdf |
    - Table 2.3.2.1. Coordinates of Patterson peaks for C₂H₆Cl₂Cu₂N₂ projection (p. 239) | html | pdf |
    - Table 2.3.2.2. Square matrix representation of vector interactions in a Patterson of a crystal with M crystallographic asymmetric units each containing N atoms (p. 239) | html | pdf |
    - Table 2.3.2.3. Position of Harker sections within a Patterson (p. 240) | html | pdf |
    - Table 2.3.5.1. Possible types of vector searches (p. 250) | html | pdf |
    - Table 2.3.5.2. Orientation of the glyceraldehyde-3-phosphate dehydrogenase molecular twofold axis in the orthorhombic cell (p. 250) | html | pdf |
    - Table 2.3.6.1. Different types of uses for the rotation function (p. 251) | html | pdf |
    - Table 2.3.6.2. Eulerian symmetry elements for all possible types of space-group rotations (p. 254) | html | pdf |
    - Table 2.3.6.3. Numbering of the rotation function space groups (p. 254) | html | pdf |
    - Table 2.3.6.4. Rotation function Eulerian space groups (pp. 256-257) | html | pdf |
    - Table 2.3.8.1. Molecular replacement: phase refinement as an iterative process (p. 261) | html | pdf |
- 2.4. Isomorphous replacement and anomalous scattering (pp. 264-275) | html | pdf | chapter contents |
  M. Vijayan and S. Ramaseshan
  - 2.4.1. Introduction (p. 264) | html | pdf |
  - 2.4.2. Isomorphous replacement method (pp. 264-265) | html | pdf |
    - 2.4.2.1. Isomorphous replacement and isomorphous addition (pp. 264-265) | html | pdf |
    - 2.4.2.2. Single isomorphous replacement method (p. 265) | html | pdf |
    - 2.4.2.3. Multiple isomorphous replacement method (p. 265) | html | pdf |
  - 2.4.3. Anomalous-scattering method (pp. 265-269) | html | pdf |
    - 2.4.3.1. Dispersion correction (pp. 265-266) | html | pdf |
    - 2.4.3.2. Violation of Friedel's law (pp. 266-267) | html | pdf |
    - 2.4.3.3. Friedel and Bijvoet pairs (p. 267) | html | pdf |
    - 2.4.3.4. Determination of absolute configuration (pp. 267-268) | html | pdf |
    - 2.4.3.5. Determination of phase angles (p. 268) | html | pdf |
    - 2.4.3.6. Anomalous scattering without phase change (p. 268) | html | pdf |
    - 2.4.3.7. Treatment of anomalous scattering in structure refinement (pp. 268-269) | html | pdf |
  - 2.4.4. Isomorphous replacement and anomalous scattering in protein crystallography (pp. 269-274) | html | pdf |
    - 2.4.4.1. Protein heavy-atom derivatives (p. 269) | html | pdf |
    - 2.4.4.2. Determination of heavy-atom parameters (pp. 269-270) | html | pdf |
    - 2.4.4.3. Refinement of heavy-atom parameters (pp. 270-271) | html | pdf |
    - 2.4.4.4. Treatment of errors in phase evaluation: Blow and Crick formulation (pp. 271-272) | html | pdf |
    - 2.4.4.5. Use of anomalous scattering in phase evaluation (pp. 272-273) | html | pdf |
    - 2.4.4.6. Estimation of r.m.s. error (pp. 273-274) | html | pdf |
    - 2.4.4.7. Suggested modifications to Blow and Crick formulation and the inclusion of phase information from other sources (p. 274) | html | pdf |
    - 2.4.4.8. Fourier representation of anomalous scatterers (p. 274) | html | pdf |
  - 2.4.5. Anomalous scattering of neutrons and synchrotron radiation. The multiwavelength method (pp. 274-275) | html | pdf |
    - 2.4.5.1. Neutron anomalous scattering (p. 275) | html | pdf |
    - 2.4.5.2. Anomalous scattering of synchrotron radiation (p. 275) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 2.4.2.1. Vector relationship between $[{\bf F}_{N}]$ and $[{\bf F}_{M}\ (\equiv {\bf F}_{NH})]$ (p. 265) | html | pdf |
    - Fig. 2.4.2.2. Relationship between $[\alpha_{N},\ \alpha_{H}]$ and $[\varphi]$ (p. 265) | html | pdf |
    - Fig. 2.4.2.3. Harker construction when two heavy-atom derivatives are available (p. 266) | html | pdf |
    - Fig. 2.4.3.1. Variation of (a) and (b) as a function of atomic number for Cu $[K\alpha]$ and Mo $[K\alpha]$ radiations (p. 266) | html | pdf |
    - Fig. 2.4.3.2. Vector diagram illustrating the violation of Friedel's law when $[{\bf F}''_{Q} \neq 0]$ (p. 267) | html | pdf |
    - Fig. 2.4.3.3. A composite view of the vector relationship between $[{\bf F}_{N}(+)]$ and $[{\bf F}_{N}(-)]$ (p. 267) | html | pdf |
    - Fig. 2.4.4.1. Distribution of intersections in the Harker construction under non-ideal conditions (p. 272) | html | pdf |
    - Fig. 2.4.4.2. Vector diagram indicating the calculated structure factor, $[{\bf D}_{Hi}(\alpha)]$ , of the ith heavy-atom derivative (p. 272) | html | pdf |
    - Fig. 2.4.4.3. The probability distribution of the protein phase angle (p. 272) | html | pdf |
    - Fig. 2.4.4.4. Harker construction using anomalous-scattering data from a single derivative (p. 273) | html | pdf |
  - Tables
    - Table 2.4.3.1. Phase angles of different components of the structure factor in space group (p. 267) | html | pdf |
- 2.5. Electron diffraction and electron microscopy in structure determination (pp. 276-345) | html | pdf | chapter contents |
  J. M. Cowley, P. Goodman, B. K. Vainshtein, B. B. Zvyagin and D. L. Dorset
  - 2.5.1. Foreword (pp. 276-277) | html | pdf |
    J. M. Cowley
  - 2.5.2. Electron diffraction and electron microscopy (pp. 277-285) | html | pdf |
    J. M. Cowley
    - 2.5.2.1. Introduction (pp. 277-278) | html | pdf |
    - 2.5.2.2. The interactions of electrons with matter (pp. 278-279) | html | pdf |
    - 2.5.2.3. Recommended sign conventions (pp. 279-280) | html | pdf |
    - 2.5.2.4. Scattering of electrons by crystals; approximations (pp. 280-281) | html | pdf |
    - 2.5.2.5. Kinematical diffraction formulae (pp. 281-282) | html | pdf |
    - 2.5.2.6. Imaging with electrons (pp. 282-283) | html | pdf |
    - 2.5.2.7. Imaging of very thin and weakly scattering objects (pp. 283-284) | html | pdf |
    - 2.5.2.8. Crystal structure imaging (p. 284) | html | pdf |
    - 2.5.2.9. Image resolution (pp. 284-285) | html | pdf |
    - 2.5.2.10. Electron diffraction in electron microscopes (p. 285) | html | pdf |
  - 2.5.3. Space-group determination by convergent-beam electron diffraction (pp. 285-306) | html | pdf |
    P. Goodman
    - 2.5.3.1. Introduction (pp. 285-286) | html | pdf |
      - 2.5.3.1.1. CBED (pp. 285-286) | html | pdf |
      - 2.5.3.1.2. Zone-axis patterns from CBED (p. 286) | html | pdf |
    - 2.5.3.2. Background theory and analytical approach (pp. 286-288) | html | pdf |
      - 2.5.3.2.1. Direct and reciprocity symmetries: types I and II (pp. 286-287) | html | pdf |
      - 2.5.3.2.2. Reciprocity and Friedel's law (p. 287) | html | pdf |
      - 2.5.3.2.3. In-disc symmetries (pp. 287-288) | html | pdf |
      - 2.5.3.2.4. Zero-layer absences (p. 288) | html | pdf |
    - 2.5.3.3. Pattern observation of individual symmetry elements (pp. 288-289) | html | pdf |
    - 2.5.3.4. Auxiliary tables (pp. 289-291) | html | pdf |
    - 2.5.3.5. Space-group analyses of single crystals; experimental procedure and published examples (pp. 291-292) | html | pdf |
      - 2.5.3.5.1. Stages of procedure (pp. 291-292) | html | pdf |
      - 2.5.3.5.2. Examples (p. 292) | html | pdf |
    - 2.5.3.6. Use of CBED in study of crystal defects, twins and non-classical crystallography (pp. 292-295) | html | pdf |
    - 2.5.3.7. Present limitations and general conclusions (p. 295) | html | pdf |
    - 2.5.3.8. Computer programs available (pp. 295-306) | html | pdf |
  - 2.5.4. Electron-diffraction structure analysis (EDSA) (pp. 306-310) | html | pdf |
    B. K. Vainshtein and B. B. Zvyagin
    - 2.5.4.1. Introduction (p. 306) | html | pdf |
    - 2.5.4.2. The geometry of ED patterns (pp. 306-308) | html | pdf |
    - 2.5.4.3. Intensities of diffraction beams (pp. 308-309) | html | pdf |
    - 2.5.4.4. Structure analysis (pp. 309-310) | html | pdf |
  - 2.5.5. Image reconstruction (pp. 310-315) | html | pdf |
    B. K. Vainshtein
    - 2.5.5.1. Introduction (pp. 310-311) | html | pdf |
    - 2.5.5.2. Thin weak phase objects at optimal defocus (pp. 311-312) | html | pdf |
    - 2.5.5.3. An account of absorption (p. 312) | html | pdf |
    - 2.5.5.4. Thick crystals (p. 312) | html | pdf |
    - 2.5.5.5. Image enhancement (pp. 313-315) | html | pdf |
  - 2.5.6. Three-dimensional reconstruction (pp. 316-320) | html | pdf |
    B. K. Vainshtein
    - 2.5.6.1. The object and its projection (pp. 315-316) | html | pdf |
    - 2.5.6.2. Orthoaxial projection (pp. 316-317) | html | pdf |
    - 2.5.6.3. Discretization (p. 317) | html | pdf |
    - 2.5.6.4. Methods of direct reconstruction (p. 317) | html | pdf |
    - 2.5.6.5. The method of back-projection (p. 318) | html | pdf |
    - 2.5.6.6. The algebraic and iteration methods (p. 318) | html | pdf |
    - 2.5.6.7. Reconstruction using Fourier transformation (pp. 318-319) | html | pdf |
    - 2.5.6.8. Three-dimensional reconstruction in the general case (pp. 319-320) | html | pdf |
  - 2.5.7. Direct phase determination in electron crystallography (pp. 320-326) | html | pdf |
    D. L. Dorset
    - 2.5.7.1. Problems with `traditional' phasing techniques (pp. 320-321) | html | pdf |
    - 2.5.7.2. Direct phase determination from electron micrographs (pp. 321-322) | html | pdf |
    - 2.5.7.3. Probabilistic estimate of phase invariant sums (pp. 322-323) | html | pdf |
    - 2.5.7.4. The tangent formula (pp. 323-324) | html | pdf |
    - 2.5.7.5. Density modification (p. 324) | html | pdf |
    - 2.5.7.6. Convolution techniques (p. 324) | html | pdf |
    - 2.5.7.7. Maximum entropy and likelihood (p. 325) | html | pdf |
    - 2.5.7.8. Influence of multiple scattering on direct electron crystallographic structure analysis (pp. 325-326) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 2.5.2.1. The variation with accelerating voltage of electrons of (a) the wavelength, λ and (b) the quantity $[\lambda [1 + (h^{2} / m_{0}^{2} c^{2} \lambda^{2})] = \lambda_{c} / \beta]$ (p. 279) | html | pdf |
    - Fig. 2.5.2.2. Diagram representing the critical components of a conventional transmission electron microscope (TEM) and a scanning transmission electron microscope (STEM) (p. 282) | html | pdf |
    - Fig. 2.5.2.3. The functions $[\chi (U)]$ , the phase factor for the transfer function of a lens given by equation (2.5.2.33), and $[\sin \chi (U)]$ for the Scherzer optimum defocus condition (p. 283) | html | pdf |
    - Fig. 2.5.3.1. Vector diagram in semi-reciprocal space, using Ewald-sphere constructions to show the `incident', `reciprocity' and `reciprocity × centrosymmetry' sets of vectors (p. 287) | html | pdf |
    - Fig. 2.5.3.2. Diagrammatic representation of a CBED disc with symmetry lines m, m_R (alternate labels G, S) and the central point $[{\bf K}_{g} = 0]$ (p. 287) | html | pdf |
    - Fig. 2.5.3.3. Diagrammatic representation of the influence of non-symmorphic elements (p. 289) | html | pdf |
    - Fig. 2.5.3.4. (a) Zone-axis patterns from hexagonal structures β- $[\hbox{Si}_{3}\hbox{N}_{4}]$ (left) and β-GaS (right) together with the appropriate planar figures for point symmetries 6 and 6mm, respectively (p. 293) | html | pdf |
    - Fig. 2.5.3.5. Zone-axis patterns from cubic structures gahnite (left) (Ishizuka & Taftø, 1982) and χ-phase precipitate (right) (Steeds & Evans, 1980) (p. 294) | html | pdf |
    - Fig. 2.5.3.6. (a) CBED pattern from the exact [001] (cyclic) zone-axis orientation of FeS₂ (p. 294) | html | pdf |
    - Fig. 2.5.4.1. Ewald spheres in reciprocal space (p. 306) | html | pdf |
    - Fig. 2.5.4.2. Triclinic reciprocal lattice (p. 307) | html | pdf |
    - Fig. 2.5.4.3. Formation of ellipses on an electron-diffraction pattern from an oblique texture (p. 307) | html | pdf |
    - Fig. 2.5.5.1. The χ function and two components of the Scherzer phase function sin $[\chi (U)]$ and cos $[\chi (U)]$ (p. 311) | html | pdf |
    - Fig. 2.5.5.2. (a) Diagram of an optical diffractometer (p. 314) | html | pdf |
    - Fig. 2.5.6.1. A three-dimensional object $[\varphi_{3}]$ and its two-dimensional projection $[\varphi_{2}]$ (p. 316) | html | pdf |
    - Fig. 2.5.6.2. The projection sphere and projection $[\varphi_{2}]$ of $[\varphi_{3}]$ along τ onto the plane $[{\bf x} \perp {\bf \boldtau}]$ (p. 316) | html | pdf |
    - Fig. 2.5.6.3. Orthoaxial projection (p. 317) | html | pdf |
    - Fig. 2.5.6.4. Discretization and oblique projection (p. 317) | html | pdf |
    - Fig. 2.5.6.5. (a) Formation of a projection function; (b) superposition of these functions (p. 318) | html | pdf |
    - Fig. 2.5.6.6. Relative position of the particle and planes of projection (p. 320) | html | pdf |
    - Fig. 2.5.6.7. Section of a three-dimensional Fourier transform of the density of the particles, corresponding to plane projections of this density (p. 320) | html | pdf |
    - Fig. 2.5.6.8. Plane projections of a three-dimensional body (p. 320) | html | pdf |
    - Fig. 2.5.7.1. Potential map for diketopiperazine (p. 323) | html | pdf |
  - Tables
    - Table 2.5.2.1. Standard crystallographic and alternative crystallographic sign conventions for electron diffraction (p. 280) | html | pdf |
    - Table 2.5.3.1. Listing of the symmetry elements relating to CBED patterns under the classifications of `vertical' (I), `horizontal' (II) and combined or roto-inversionary axes (p. 286) | html | pdf |
    - Table 2.5.3.2. Diagrammatic illustrations of the actions of five types of symmetry elements (given in the last column in Volume A diagrammatic symbols) on an asymmetric pattern component, in relation to the centre of the pattern at $[{\bf K}_{00} = {\bf 0}]$ , shown as `⊕', or in relation to the centre of a diffraction order at $[{\bf K}_{0g} = {\bf 0}]$ , shown as `+' (p. 288) | html | pdf |
    - Table 2.5.3.3. Diffraction point-group tables, giving whole-pattern and central-beam pattern symmetries in terms of BESR diffraction-group symbols and diperiodic group symbols (p. 290) | html | pdf |
    - Table 2.5.3.4. Tabulation of principal-axis CBED pattern symmetries against relevant space groups given as numbers (pp. 296-297) | html | pdf |
    - Table 2.5.3.5. Conditions for observation of GS bands for the 137 space groups exhibiting these extinctions (pp. 298-305) | html | pdf |

Dual bases in crystallographic computing
- 3.1. Distances, angles, and their standard uncertainties (pp. 348-352) | html | pdf | chapter contents |
  D. E. Sands
  - 3.1.1. Introduction (p. 348) | html | pdf |
  - 3.1.2. Scalar product (p. 348) | html | pdf |
  - 3.1.3. Length of a vector (p. 348) | html | pdf |
  - 3.1.4. Angle between two vectors (p. 348) | html | pdf |
  - 3.1.5. Vector product (p. 349) | html | pdf |
  - 3.1.6. Permutation tensors (p. 349) | html | pdf |
  - 3.1.7. Components of vector product (p. 349) | html | pdf |
  - 3.1.8. Some vector relationships (p. 349) | html | pdf |
    - 3.1.8.1. Triple vector product (p. 349) | html | pdf |
    - 3.1.8.2. Scalar product of vector products (p. 349) | html | pdf |
    - 3.1.8.3. Vector product of vector products (p. 349) | html | pdf |
  - 3.1.9. Planes (pp. 349-350) | html | pdf |
  - 3.1.10. Variance–covariance matrices (pp. 350-351) | html | pdf |
  - 3.1.11. Mean values (pp. 351-352) | html | pdf |
  - 3.1.12. Computation (p. 352) | html | pdf |
  - References | html | pdf |
- 3.2. The least-squares plane (pp. 353-359) | html | pdf | chapter contents |
  R. E. Marsh and V. Schomaker
  - 3.2.1. Introduction (p. 353) | html | pdf |
  - 3.2.2. Least-squares plane based on uncorrelated, isotropic weights (pp. 353-355) | html | pdf |
    - 3.2.2.1. Error propagation (pp. 353-355) | html | pdf |
    - 3.2.2.2. The standard uncertainty of the distance from an atom to the plane (p. 355) | html | pdf |
  - 3.2.3. The proper least-squares plane, with Gaussian weights (pp. 355-358) | html | pdf |
    - 3.2.3.1. Formulation and solution of the general Gaussian plane (pp. 356-358) | html | pdf |
    - 3.2.3.2. Concluding remarks (p. 358) | html | pdf |
  - Appendix 3.2.1. (pp. 358-359) | html | pdf |
  - References | html | pdf |
- 3.3. Molecular modelling and graphics (pp. 360-384) | html | pdf | chapter contents |
  R. Diamond
  - 3.3.1. Graphics (pp. 360-376) | html | pdf |
    - 3.3.1.1. Coordinate systems, notation and standards (pp. 360-361) | html | pdf |
      - 3.3.1.1.1. Cartesian and crystallographic coordinates (p. 360) | html | pdf |
      - 3.3.1.1.2. Homogeneous coordinates (pp. 360-361) | html | pdf |
      - 3.3.1.1.3. Notation (p. 361) | html | pdf |
      - 3.3.1.1.4. Standards (p. 361) | html | pdf |
    - 3.3.1.2. Orthogonal (or rotation) matrices (pp. 361-367) | html | pdf |
      - 3.3.1.2.1. General form (pp. 361-363) | html | pdf |
      - 3.3.1.2.2. Measurement of rotations and strains from coordinates (pp. 364-366) | html | pdf |
      - 3.3.1.2.3. Orthogonalization of impure rotations (p. 367) | html | pdf |
      - 3.3.1.2.4. Eigenvalues and eigenvectors of orthogonal matrices (p. 367) | html | pdf |
    - 3.3.1.3. Projection transformations and spaces (pp. 367-373) | html | pdf |
      - 3.3.1.3.1. Definitions (pp. 367-368) | html | pdf |
      - 3.3.1.3.2. Translation (p. 368) | html | pdf |
      - 3.3.1.3.3. Rotation (p. 368) | html | pdf |
      - 3.3.1.3.4. Scale (p. 368) | html | pdf |
      - 3.3.1.3.5. Windowing and perspective (pp. 368-370) | html | pdf |
      - 3.3.1.3.6. Stereoviews (p. 370) | html | pdf |
      - 3.3.1.3.7. Viewports (pp. 370-371) | html | pdf |
      - 3.3.1.3.8. Compound transformations (pp. 371-372) | html | pdf |
      - 3.3.1.3.9. Inverse transformations (p. 372) | html | pdf |
      - 3.3.1.3.10. The three-axis joystick (pp. 372-373) | html | pdf |
      - 3.3.1.3.11. Other useful rotations (p. 373) | html | pdf |
      - 3.3.1.3.12. Symmetry (p. 373) | html | pdf |
    - 3.3.1.4. Modelling transformations (pp. 373-374) | html | pdf |
      - 3.3.1.4.1. Rotation about a bond (p. 373) | html | pdf |
      - 3.3.1.4.2. Stacked transformations (pp. 373-374) | html | pdf |
    - 3.3.1.5. Drawing techniques (pp. 374-376) | html | pdf |
      - 3.3.1.5.1. Types of hardware (pp. 374-375) | html | pdf |
      - 3.3.1.5.2. Optimization of line drawings (p. 375) | html | pdf |
      - 3.3.1.5.3. Representation of surfaces by lines (p. 375) | html | pdf |
      - 3.3.1.5.4. Representation of surfaces by dots (p. 375) | html | pdf |
      - 3.3.1.5.5. Representation of surfaces by shading (pp. 375-376) | html | pdf |
      - 3.3.1.5.6. Advanced hidden-line and hidden-surface algorithms (p. 376) | html | pdf |
  - 3.3.2. Molecular modelling, problems and approaches (pp. 377-380) | html | pdf |
    - 3.3.2.1. Connectivity (p. 377) | html | pdf |
      - 3.3.2.1.1. Connectivity tables (p. 377) | html | pdf |
      - 3.3.2.1.2. Implied connectivity (p. 377) | html | pdf |
    - 3.3.2.2. Modelling methods (pp. 377-380) | html | pdf |
      - 3.3.2.2.1. Methods based on conformational variables (pp. 378-379) | html | pdf |
      - 3.3.2.2.2. Methods based on positional coordinates (p. 379) | html | pdf |
      - 3.3.2.2.3. Approaches to the problem of multiple minima (pp. 379-380) | html | pdf |
  - 3.3.3. Implementations (pp. 380-384) | html | pdf |
    - 3.3.3.1. Systems for the display and modification of retrieved data (pp. 380-381) | html | pdf |
      - 3.3.3.1.1. ORTEP (p. 380) | html | pdf |
      - 3.3.3.1.2. Feldmann's system (pp. 380-381) | html | pdf |
      - 3.3.3.1.3. Lesk & Hardman software (p. 381) | html | pdf |
      - 3.3.3.1.4. GRAMPS (p. 381) | html | pdf |
      - 3.3.3.1.5. Takenaka & Sasada's system (p. 381) | html | pdf |
      - 3.3.3.1.6. MIDAS (p. 381) | html | pdf |
      - 3.3.3.1.7. Insight (p. 381) | html | pdf |
      - 3.3.3.1.8. PLUTO (p. 381) | html | pdf |
      - 3.3.3.1.9. MDKINO (p. 381) | html | pdf |
    - 3.3.3.2. Molecular-modelling systems based on electron density (pp. 381-384) | html | pdf |
      - 3.3.3.2.1. CHEMGRAF (pp. 381-382) | html | pdf |
      - 3.3.3.2.2. GRIP (p. 382) | html | pdf |
      - 3.3.3.2.3. Barry, Denson & North's systems (p. 382) | html | pdf |
      - 3.3.3.2.4. MMS-X (p. 382) | html | pdf |
      - 3.3.3.2.5. Texas A&M University system (p. 382) | html | pdf |
      - 3.3.3.2.6. Bilder (pp. 382-383) | html | pdf |
      - 3.3.3.2.7. Frodo (p. 383) | html | pdf |
      - 3.3.3.2.8. Guide (p. 383) | html | pdf |
      - 3.3.3.2.9. HYDRA (p. 383) | html | pdf |
      - 3.3.3.2.10. O (p. 384) | html | pdf |
    - 3.3.3.3. Molecular-modelling systems based on other criteria (p. 384) | html | pdf |
      - 3.3.3.3.1. Molbuild , Rings , PRXBLD and MM2/MMP2 (p. 384) | html | pdf |
      - 3.3.3.3.2. Script (p. 384) | html | pdf |
      - 3.3.3.3.3. CHARMM (p. 384) | html | pdf |
      - 3.3.3.3.4. Commercial systems (p. 384) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 3.3.1.1. The relationship between display-space coordinates (X, Y, Z, W) and picture-space coordinates (x, y, z, w) derived from them by the window transformation, U (p. 369) | html | pdf |
    - Fig. 3.3.1.2. Schematic representation of a simple branched-chain molecule with a stationary root and two extremities (p. 374) | html | pdf |
- 3.4. Accelerated convergence treatment of R⁻ⁿ lattice sums (pp. 385-397) | html | pdf | chapter contents |
  D. E. Williams
  - 3.4.1. Introduction (p. 385) | html | pdf |
  - 3.4.2. Definition and behaviour of the direct-space sum (p. 385) | html | pdf |
  - 3.4.3. Preliminary description of the method (pp. 385-386) | html | pdf |
  - 3.4.4. Preliminary derivation to obtain a formula which accelerates the convergence of an $[{\bi R}^{\bf -{\bi n}}]$ sum over lattice points X(d) (pp. 386-388) | html | pdf |
  - 3.4.5. Extension of the method to a composite lattice (pp. 388-389) | html | pdf |
  - 3.4.6. The case of $[{\bf{\bi n} = 1}]$ (Coulombic lattice energy) (p. 389) | html | pdf |
  - 3.4.7. The cases of $[{\bf{\bi n} = 2}]$ and $[{\bf{\bi n} = 3}]$ (p. 389) | html | pdf |
  - 3.4.8. Derivation of the accelerated convergence formula via the Patterson function (pp. 389-390) | html | pdf |
  - 3.4.9. Evaluation of the incomplete gamma function (p. 390) | html | pdf |
  - 3.4.10. Summation over the asymmetric unit and elimination of intramolecular energy terms (p. 390) | html | pdf |
  - 3.4.11. Reference formulae for particular values of n (pp. 390-391) | html | pdf |
  - 3.4.12. Numerical illustrations (pp. 391-392) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 3.4.2.1. Untreated lattice-sum results for the Coulombic energy () of sodium chloride (kJ mol⁻¹, Å); the lattice constant is taken as 5.628 Å (p. 385) | html | pdf |
    - Table 3.4.2.2. Untreated lattice-sum results for the dispersion energy () of crystalline benzene (kJ mol⁻¹, Å) (p. 386) | html | pdf |
    - Table 3.4.12.1. Accelerated-convergence results for the Coulombic sum () of sodium chloride (kJ mol⁻¹, Å): the direct sum plus the constant term (p. 391) | html | pdf |
    - Table 3.4.12.2. The reciprocal-lattice results (kJ mol⁻¹, Å) for the Coulombic sum () of sodium chloride (p. 392) | html | pdf |
    - Table 3.4.12.3. Accelerated-convergence results for the dispersion sum () of crystalline benzene (kJ mol⁻¹, Å); the figures shown are the direct-lattice sum plus the two constant terms (p. 392) | html | pdf |
    - Table 3.4.12.4. The reciprocal-lattice results (kJ mol⁻¹, Å) for the dispersion sum () of crystalline benzene (p. 392) | html | pdf |
    - Table 3.4.12.5. Approximate time (s) required to evaluate the dispersion sum () for crystalline benzene within 0.001 kJ mol⁻¹ truncation error (p. 392) | html | pdf |

Diffuse scattering and related topics
- 4.1. Thermal diffuse scattering of X-rays and neutrons (pp. 400-406) | html | pdf | chapter contents |
  B. T. M. Willis
  - 4.1.1. Introduction (p. 400) | html | pdf |
  - 4.1.2. Dynamics of three-dimensional crystals (pp. 400-402) | html | pdf |
    - 4.1.2.1. Equations of motion (pp. 401-402) | html | pdf |
    - 4.1.2.2. Quantization of normal modes. Phonons (p. 402) | html | pdf |
    - 4.1.2.3. Einstein and Debye models (p. 402) | html | pdf |
    - 4.1.2.4. Molecular crystals (p. 402) | html | pdf |
  - 4.1.3. Scattering of X-rays by thermal vibrations (pp. 403-404) | html | pdf |
  - 4.1.4. Scattering of neutrons by thermal vibrations (pp. 404-405) | html | pdf |
  - 4.1.5. Phonon dispersion relations (pp. 405-406) | html | pdf |
    - 4.1.5.1. Measurement with X-rays (p. 405) | html | pdf |
    - 4.1.5.2. Measurement with neutrons (p. 405) | html | pdf |
    - 4.1.5.3. Interpretation of dispersion relations (pp. 405-406) | html | pdf |
  - 4.1.6. Measurement of elastic constants (p. 406) | html | pdf |
  - References | html | pdf |
- 4.2. Disorder diffuse scattering of X-rays and neutrons (pp. 407-442) | html | pdf | chapter contents |
  H. Jagodzinski and F. Frey
  - 4.2.1. Scope of this chapter (pp. 407-408) | html | pdf |
  - 4.2.2. Summary of basic scattering theory (pp. 408-410) | html | pdf |
  - 4.2.3. General treatment (pp. 410-420) | html | pdf |
    - 4.2.3.1. Qualitative interpretation of diffuse scattering (pp. 410-418) | html | pdf |
      - 4.2.3.1.1. Fourier transforms (pp. 410-411) | html | pdf |
      - 4.2.3.1.2. Applications (pp. 411-418) | html | pdf |
    - 4.2.3.2. Guideline to solve a disorder problem (pp. 418-420) | html | pdf |
  - 4.2.4. Quantitative interpretation (pp. 420-438) | html | pdf |
    - 4.2.4.1. Introduction (pp. 420-421) | html | pdf |
    - 4.2.4.2. One-dimensional disorder of ordered layers (pp. 421-425) | html | pdf |
      - 4.2.4.2.1. Stacking disorder in close-packed structures (pp. 423-425) | html | pdf |
    - 4.2.4.3. Two-dimensional disorder of chains (pp. 425-429) | html | pdf |
      - 4.2.4.3.1. Scattering by randomly distributed collinear chains (pp. 425-427) | html | pdf |
      - 4.2.4.3.2. Disorder within randomly distributed collinear chains  (pp. 427-429) | html | pdf |
        4.2.4.3.2.1. General treatment  (p. 427) | html | pdf |
        4.2.4.3.2.2. Orientational disorder  (p. 427) | html | pdf |
        4.2.4.3.2.3. Longitudinal disorder  (pp. 428-429) | html | pdf |
      - 4.2.4.3.3. Correlations between different almost collinear chains (p. 429) | html | pdf |
    - 4.2.4.4. Disorder with three-dimensional correlations (defects, local ordering and clustering) (pp. 429-436) | html | pdf |
      - 4.2.4.4.1. General formulation (elastic diffuse scattering) (pp. 429-431) | html | pdf |
      - 4.2.4.4.2. Random distribution (pp. 431-432) | html | pdf |
      - 4.2.4.4.3. Short-range order in multi-component systems (p. 432) | html | pdf |
      - 4.2.4.4.4. Displacements: general remarks (pp. 432-433) | html | pdf |
      - 4.2.4.4.5. Distortions in binary systems (pp. 433-435) | html | pdf |
      - 4.2.4.4.6. Powder diffraction (p. 435) | html | pdf |
      - 4.2.4.4.7. Small concentrations of defects (p. 435) | html | pdf |
      - 4.2.4.4.8. Cluster method (p. 435) | html | pdf |
      - 4.2.4.4.9. Comparison between X-ray and neutron methods (pp. 435-436) | html | pdf |
      - 4.2.4.4.10. Dynamic properties of defects (p. 436) | html | pdf |
    - 4.2.4.5. Orientational disorder (pp. 436-438) | html | pdf |
      - 4.2.4.5.1. General expressions (pp. 436-437) | html | pdf |
      - 4.2.4.5.2. Rotational structure (form) factor (pp. 437-438) | html | pdf |
      - 4.2.4.5.3. Short-range correlations (p. 438) | html | pdf |
  - 4.2.5. Measurement of diffuse scattering (pp. 438-442) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 4.2.3.1. Model of the two-dimensional distribution of point defects, causing changes in the surroundings (p. 412) | html | pdf |
    - Fig. 4.2.3.2. One-dimensional Patterson functions of various point-defect distributions (p. 412) | html | pdf |
    - Fig. 4.2.3.3. Periodic array of domains consisting of two different atoms, represented by different heights (p. 413) | html | pdf |
    - Fig. 4.2.3.4. Influence of distortions at the boundary of domains, and separation into two parts (p. 415) | html | pdf |
    - Fig. 4.2.3.5. Typical distributions of mixed crystals (unmixing) (p. 416) | html | pdf |
    - Fig. 4.2.3.6. Decomposition of Fig. 4.2.3.5(a) into a periodic and a rapidly convergent part (p. 416) | html | pdf |
    - Fig. 4.2.3.7. The same distribution (cf. Fig. 4.2.3.5) in the case of superstructure formation (p. 417) | html | pdf |
    - Fig. 4.2.4.1. Construction of the correlation function in the method of overlapping clusters (p. 433) | html | pdf |
    - Fig. 4.2.5.1. Schematic sketch of a diffractometer setting (p. 440) | html | pdf |
    - Fig. 4.2.5.2. Line profiles in powder diffraction for sharp and diffuse reflections (p. 442) | html | pdf |
- 4.3. Diffuse scattering in electron diffraction (pp. 443-448) | html | pdf | chapter contents |
  J. M. Cowley and J. K. Gjønnes
  - 4.3.1. Introduction (pp. 443-444) | html | pdf |
  - 4.3.2. Inelastic scattering (p. 444) | html | pdf |
  - 4.3.3. Kinematical and pseudo-kinematical scattering (p. 445) | html | pdf |
  - 4.3.4. Dynamical scattering: Bragg scattering effects (pp. 445-447) | html | pdf |
  - 4.3.5. Multislice calculations for diffraction and imaging (p. 447) | html | pdf |
  - 4.3.6. Qualitative interpretation of diffuse scattering of electrons (pp. 447-448) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 4.3.1.1. Comparison between the kinematical inelastic scattering (full line) and elastic scattering (broken) for electrons and X-rays (p. 443) | html | pdf |
    - Fig. 4.3.1.2. Electron-diffraction pattern from a disordered crystal of 17Nb₂O₅·48WO₃ (p. 444) | html | pdf |
- 4.4. Scattering from mesomorphic structures (pp. 449-465) | html | pdf | chapter contents |
  P. S. Pershan
  - 4.4.1. Introduction (pp. 449-451) | html | pdf |
  - 4.4.2. The nematic phase (pp. 451-453) | html | pdf |
  - 4.4.3. Smectic-A and smectic-C phases (pp. 453-456) | html | pdf |
    - 4.4.3.1. Homogeneous smectic-A and smectic-C phases (pp. 453-455) | html | pdf |
    - 4.4.3.2. Modulated smectic-A and smectic-C phases (p. 455) | html | pdf |
    - 4.4.3.3. Surface effects (pp. 455-456) | html | pdf |
  - 4.4.4. Phases with in-plane order (pp. 456-463) | html | pdf |
    - 4.4.4.1. Hexatic phases in two dimensions (pp. 457-458) | html | pdf |
    - 4.4.4.2. Hexatic phases in three dimensions (pp. 458-460) | html | pdf |
      - 4.4.4.2.1. Hexatic-B (p. 458) | html | pdf |
      - 4.4.4.2.2. Smectic-F, smectic-I (pp. 458-460) | html | pdf |
    - 4.4.4.3. Crystalline phases with molecular rotation (pp. 460-462) | html | pdf |
      - 4.4.4.3.1. Crystal-B (pp. 460-462) | html | pdf |
      - 4.4.4.3.2. Crystal-G, crystal-J (p. 462) | html | pdf |
    - 4.4.4.4. Crystalline phases with herringbone packing (pp. 462-463) | html | pdf |
      - 4.4.4.4.1. Crystal-E (pp. 462-463) | html | pdf |
      - 4.4.4.4.2. Crystal-H, crystal-K (p. 463) | html | pdf |
  - 4.4.5. Discotic phases (p. 463) | html | pdf |
  - 4.4.6. Other phases (pp. 463-464) | html | pdf |
  - 4.4.7. Notes added in proof to first edition (pp. 464-465) | html | pdf |
    - 4.4.7.1. Phases with intermediate molecular tilt: smectic-L, crystalline-M,N (p. 464) | html | pdf |
    - 4.4.7.2. Nematic to smectic-A phase transition (pp. 464-465) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 4.4.1.1. Illustration of the progression of order throughout the sequence of mesomorphic phases that are based on `rod-like' molecules (p. 449) | html | pdf |
    - Fig. 4.4.1.2. Schematic illustration of the real-space molecular order and the scattering cross sections in reciprocal space for the: (a) nematic; (b) smectic-A; and (c), (d) smectic-C phases (p. 450) | html | pdf |
    - Fig. 4.4.1.3. Chemical formulae for some of the molecules that form thermotropic liquid crystals (p. 451) | html | pdf |
    - Fig. 4.4.3.1. (a) Schematic illustration of the necessary condition for coupling between order parameters when $[|{\bf q}_{2}|\;\lt\; 2|{\bf q}_{1}|]$ ; $[|{\bf q}| = (|{\bf q}_{2}|^{2} - |{\bf q}_{1}|^{2})^{1/2} = |{\bf q}_{1}| \sin (\alpha)]$ (p. 455) | html | pdf |
    - Fig. 4.4.3.2. Specular reflectivity of ∼8 keV X-rays from the air–liquid interface of the nematic liquid crystal 8OCB 0.05 K above the nematic to smectic-A transition temperature (p. 456) | html | pdf |
    - Fig. 4.4.4.1. (a) Schematic illustration of the geometry and (b) kinematics of X-ray scattering from a freely suspended smectic film (p. 457) | html | pdf |
    - Fig. 4.4.4.2. Typical $[Q_{L}]$ scans from the crystalline-B phases of (a) a free film of 7O.7 (p. 457) | html | pdf |
    - Fig. 4.4.4.3. Scattering intensities in reciprocal space from two-dimensional: (a) liquid; (b) crystal; (c) normal hexatic; and tilted hexatics (p. 458) | html | pdf |
    - Fig. 4.4.4.4. Scattering intensities in reciprocal space from three-dimensional tilted hexatic phases (p. 459) | html | pdf |
    - Fig. 4.4.4.5. The phase diagram for free films of 7O.7 as a function of thickness and temperature (p. 459) | html | pdf |
    - Fig. 4.4.4.6. Location of the Bragg peaks in one 60° section of reciprocal space for the three-dimensional crystalline-B phases observed in thick films of 7O.7 (p. 461) | html | pdf |
    - Fig. 4.4.4.7. (a) The `herringbone' stacking suggested for the crystalline-E phase in which molecular rotation is partially restricted (p. 462) | html | pdf |
    - Fig. 4.4.5.1. Schematic illustration of the molecular stacking for the discotic (a) $[D_{2}]$ and (b) $[D_{1}]$ phases (p. 463) | html | pdf |
  - Tables
    - Table 4.4.1.1. Some of the symmetry properties of the series of three-dimensional phases described in Fig. 4.4.1.1 (p. 449) | html | pdf |
    - Table 4.4.1.2. The symmetry properties of the two-dimensional hexatic and crystalline phases (p. 450) | html | pdf |
    - Table 4.4.2.1. Summary of critical exponents from X-ray scattering studies of the nematic to smectic-A phase transition (p. 453) | html | pdf |
- 4.5. Polymer crystallography (pp. 466-485) | html | pdf | chapter contents |
  R. P. Millane and D. L. Dorset
  - 4.5.1. Overview (p. 466) | html | pdf |
    R. P. Millane and D. L. Dorset
  - 4.5.2. X-ray fibre diffraction analysis (pp. 466-481) | html | pdf |
    R. P. Millane
    - 4.5.2.1. Introduction (pp. 466-467) | html | pdf |
    - 4.5.2.2. Fibre specimens (p. 467) | html | pdf |
    - 4.5.2.3. Diffraction by helical structures (pp. 467-469) | html | pdf |
      - 4.5.2.3.1. Helix symmetry (pp. 467-468) | html | pdf |
      - 4.5.2.3.2. Diffraction by helical structures (pp. 468-469) | html | pdf |
      - 4.5.2.3.3. Approximate helix symmetry (p. 469) | html | pdf |
    - 4.5.2.4. Diffraction by fibres (pp. 469-472) | html | pdf |
      - 4.5.2.4.1. Noncrystalline fibres (p. 469) | html | pdf |
      - 4.5.2.4.2. Polycrystalline fibres (pp. 469-470) | html | pdf |
      - 4.5.2.4.3. Random copolymers (pp. 470-471) | html | pdf |
      - 4.5.2.4.4. Partially crystalline fibres (pp. 471-472) | html | pdf |
    - 4.5.2.5. Processing diffraction data (pp. 472-474) | html | pdf |
    - 4.5.2.6. Structure determination (pp. 474-481) | html | pdf |
      - 4.5.2.6.1. Overview (pp. 474-475) | html | pdf |
      - 4.5.2.6.2. Helix symmetry, cell constants and space-group symmetry (p. 475) | html | pdf |
      - 4.5.2.6.3. Patterson functions (pp. 475-476) | html | pdf |
      - 4.5.2.6.4. Molecular model building (pp. 476-477) | html | pdf |
      - 4.5.2.6.5. Difference Fourier synthesis (pp. 477-478) | html | pdf |
      - 4.5.2.6.6. Multidimensional isomorphous replacement (pp. 478-479) | html | pdf |
      - 4.5.2.6.7. Other techniques (pp. 479-480) | html | pdf |
      - 4.5.2.6.8. Reliability (pp. 480-481) | html | pdf |
  - 4.5.3. Electron crystallography of polymers (pp. 481-485) | html | pdf |
    D. L. Dorset
    - 4.5.3.1. Is polymer electron crystallography possible? (p. 481) | html | pdf |
    - 4.5.3.2. Crystallization and data collection (pp. 481-482) | html | pdf |
    - 4.5.3.3. Crystal structure analysis (pp. 482-483) | html | pdf |
    - 4.5.3.4. Examples of crystal structure analyses (pp. 483-485) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 4.5.2.1. Fibre diffraction geometry (see text) (p. 473) | html | pdf |
    - Fig. 4.5.2.2. Flow chart of the molecular-model-building approach to structure determination (Arnott, 1980) (p. 476) | html | pdf |
    - Fig. 4.5.3.1. Initial potential map for poly-γ-methyl-L-glutamate (plane group pg) found with phases generated by the Sayre equation (p. 484) | html | pdf |
    - Fig. 4.5.3.2. Crystal structures of linear polymers determined from three-dimensional data (p. 484) | html | pdf |
  - Tables
    - Table 4.5.3.1. Structure analysis of poly-γ-methyl-L-glutamate in the β form (p. 483) | html | pdf |
- 4.6. Reciprocal-space images of aperiodic crystals (pp. 486-532) | html | pdf | chapter contents |
  W. Steurer and T. Haibach
  - 4.6.1. Introduction (pp. 486-487) | html | pdf |
  - 4.6.2. The n-dimensional description of aperiodic crystals (pp. 487-494) | html | pdf |
    - 4.6.2.1. Basic concepts (p. 487) | html | pdf |
    - 4.6.2.2. 1D incommensurately modulated structures (pp. 487-489) | html | pdf |
    - 4.6.2.3. 1D composite structures (pp. 489-490) | html | pdf |
    - 4.6.2.4. 1D quasiperiodic structures (pp. 490-493) | html | pdf |
    - 4.6.2.5. 1D structures with fractal atomic surfaces (pp. 493-494) | html | pdf |
  - 4.6.3. Reciprocal-space images (pp. 494-515) | html | pdf |
    - 4.6.3.1. Incommensurately modulated structures (IMSs) (pp. 494-497) | html | pdf |
      - 4.6.3.1.1. Indexing (p. 495) | html | pdf |
      - 4.6.3.1.2. Diffraction symmetry (pp. 495-496) | html | pdf |
      - 4.6.3.1.3. Structure factor (pp. 496-497) | html | pdf |
    - 4.6.3.2. Composite structures (CSs) (pp. 497-498) | html | pdf |
      - 4.6.3.2.1. Indexing (p. 498) | html | pdf |
      - 4.6.3.2.2. Diffraction symmetry (p. 498) | html | pdf |
      - 4.6.3.2.3. Structure factor (p. 498) | html | pdf |
    - 4.6.3.3. Quasiperiodic structures (QSs) (pp. 498-515) | html | pdf |
      - 4.6.3.3.1. 3D structures with 1D quasiperiodic order  (pp. 498-502) | html | pdf |
        4.6.3.3.1.1. Indexing  (p. 499) | html | pdf |
        4.6.3.3.1.2. Diffraction symmetry  (p. 499) | html | pdf |
        4.6.3.3.1.3. Structure factor  (pp. 500-501) | html | pdf |
        4.6.3.3.1.4. Intensity statistics  (p. 501) | html | pdf |
        4.6.3.3.1.5. Relationships between structure factors at symmetry-related points of the Fourier image  (pp. 501-502) | html | pdf |
      - 4.6.3.3.2. Decagonal phases  (pp. 503-509) | html | pdf |
        4.6.3.3.2.1. Indexing  (p. 505) | html | pdf |
        4.6.3.3.2.2. Diffraction symmetry  (pp. 505-506) | html | pdf |
        4.6.3.3.2.3. Structure factor  (pp. 506-507) | html | pdf |
        4.6.3.3.2.4. Intensity statistics  (p. 507) | html | pdf |
        4.6.3.3.2.5. Relationships between structure factors at symmetry-related points of the Fourier image  (pp. 508-509) | html | pdf |
      - 4.6.3.3.3. Icosahedral phases  (pp. 509-515) | html | pdf |
        4.6.3.3.3.1. Indexing  (pp. 511-512) | html | pdf |
        4.6.3.3.3.2. Diffraction symmetry  (p. 512) | html | pdf |
        4.6.3.3.3.3. Structure factor  (pp. 512-513) | html | pdf |
        4.6.3.3.3.4. Intensity statistics  (pp. 513-514) | html | pdf |
        4.6.3.3.3.5. Relationships between structure factors at symmetry-related points of the Fourier image  (pp. 514-515) | html | pdf |
  - 4.6.4. Experimental aspects of the reciprocal-space analysis of aperiodic crystals (pp. 516-517) | html | pdf |
    - 4.6.4.1. Data-collection strategies (pp. 516-517) | html | pdf |
    - 4.6.4.2. Commensurability versus incommensurability (p. 517) | html | pdf |
    - 4.6.4.3. Twinning and nanodomain structures (p. 517) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 4.6.2.1. The combination of a basic structure $[s({\bf r})]$ , with period a, and a sinusoidal modulation function , with amplitude A, period λ and $[t = {\bf q}\cdot {\bf r}]$ , gives a modulated structure (MS) $[s_{m}({\bf r})]$ (p. 488) | html | pdf |
    - Fig. 4.6.2.2. 2D embedding of the sinusoidally modulated structure illustrated in Fig. 4.6.2.1 (p. 488) | html | pdf |
    - Fig. 4.6.2.3. Schematic representation of the 2D reciprocal-space embedding of the 1D sinusoidally modulated structure depicted in Figs. 4.6.2.1 and 4.6.2.2 (p. 489) | html | pdf |
    - Fig. 4.6.2.4. 2D embedding of a 1D composite structure without mutual interaction of the subsystems (p. 489) | html | pdf |
    - Fig. 4.6.2.5. 2D embedding of a 1D composite structure with mutual interaction of the subsystems causing modulations (p. 490) | html | pdf |
    - Fig. 4.6.2.6. Schematic representation of the reciprocal space of the embedded 1D composite structure depicted in Fig. 4.6.2.4 (p. 490) | html | pdf |
    - Fig. 4.6.2.7. Schematic representation of the reciprocal space of the embedded 1D composite structure depicted in Fig. 4.6.2.5 (p. 490) | html | pdf |
    - Fig. 4.6.2.8. 2D embedding of the Fibonacci chain (p. 491) | html | pdf |
    - Fig. 4.6.2.9. Schematic representation of the reciprocal space of the embedded Fibonacci chain depicted in Fig. 4.6.2.8 (p. 492) | html | pdf |
    - Fig. 4.6.2.10. Reciprocal space of the embedded Fibonacci chain as a modulated structure (p. 493) | html | pdf |
    - Fig. 4.6.2.11. 2D direct-space embedding of the Fibonacci chain as a modulated structure (p. 493) | html | pdf |
    - Fig. 4.6.2.12. (a) Three steps in the development of the fractal atomic surface of the squared Fibonacci sequence starting from an initiator and a generator (p. 494) | html | pdf |
    - Fig. 4.6.3.1. Schematic diffraction patterns for IMSs with (a) 1D, (b) 2D and (c) 3D modulation (p. 494) | html | pdf |
    - Fig. 4.6.3.2. The relationships between the coordinates $[x_{1k}, x_{4k}, \bar{x}_{1}, \bar{x}_{4}]$ and the modulation function $[u_{1k}]$ in a special section of the $[(3 + d)\hbox{D}]$ space (p. 496) | html | pdf |
    - Fig. 4.6.3.3. Schematic diffraction patterns for 3D IMSs with (a) 1D harmonic and (b) rectangular density modulation (p. 497) | html | pdf |
    - Fig. 4.6.3.4. The relative structure-factor magnitudes of mth-order satellite reflections for a harmonic displacive modulation are proportional to the values of the mth-order Bessel function $[J_{m}(x)]$ (p. 498) | html | pdf |
    - Fig. 4.6.3.5. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the parallel-space component $[|{\bf H}^{\parallel}|]$ of the diffraction vector (p. 499) | html | pdf |
    - Fig. 4.6.3.6. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the perpendicular-space component $[|{\bf H}^{\perp}|]$ of the diffraction vector (p. 500) | html | pdf |
    - Fig. 4.6.3.7. The structure factors $[F({\bf H})]$ of the Fibonacci chain decorated with aluminium atoms (U_overall = 0.005 Å²) as a function of the parallel (above) and the perpendicular (below) component of the diffraction vector (p. 500) | html | pdf |
    - Fig. 4.6.3.8. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of the squared Fibonacci chain decorated with equal point atoms are shown as a function of the parallel-space component $[|{\bf H}^{\parallel}|]$ of the diffraction vector (p. 501) | html | pdf |
    - Fig. 4.6.3.9. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of the squared Fibonacci chain decorated with equal point atoms are shown as a function of the perpendicular-space component $[|{\bf H}^{\perp}|]$ of the diffraction vector (p. 502) | html | pdf |
    - Fig. 4.6.3.10. Part … LSLLSLSL … of a Fibonacci sequence $[s({\bf r})]$ before and after scaling by the factor τ (p. 502) | html | pdf |
    - Fig. 4.6.3.11. Scaling operations of the Fibonacci sequence (p. 502) | html | pdf |
    - Fig. 4.6.3.12. Reciprocal basis of the decagonal phase in the 5D description projected upon $[{\bf V}^{\parallel}]$ (above left) and $[{\bf V}^{\perp}]$ (above right) (p. 503) | html | pdf |
    - Fig. 4.6.3.13. A section of a Penrose tiling (thin lines) superposed by its τ-deflated tiling (above, thick lines) and by its $[\tau^{2}]$ -deflated tiling (below, thick lines) (p. 504) | html | pdf |
    - Fig. 4.6.3.14. Scaling symmetry of a pentagram superposed on the Penrose tiling (p. 504) | html | pdf |
    - Fig. 4.6.3.15. Atomic surface of the Penrose tiling in the 5D hypercubic description (p. 505) | html | pdf |
    - Fig. 4.6.3.16. Projection of the 4D hyper-rhombohedral unit cell of the Penrose tiling in the 4D description upon the perpendicular space (p. 505) | html | pdf |
    - Fig. 4.6.3.17. Schematic diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) (p. 506) | html | pdf |
    - Fig. 4.6.3.18. Enlarged section of Fig. 4.6.3.17 (p. 506) | html | pdf |
    - Fig. 4.6.3.19. The perpendicular-space diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) (p. 506) | html | pdf |
    - Fig. 4.6.3.20. Enlarged section of Fig. 4.6.3.19 showing the projected 4D reciprocal-lattice unit cell (p. 506) | html | pdf |
    - Fig. 4.6.3.21. Radial distribution function of the structure factors $[F({\bf H})]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\parallel}]$ (p. 508) | html | pdf |
    - Fig. 4.6.3.22. Radial distribution function of the structure factors $[F({\bf H})]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\perp}]$ (p. 508) | html | pdf |
    - Fig. 4.6.3.23. Radial distribution function of the structure-factor magnitudes $[|F({\bf H})|]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\perp}]$ (p. 508) | html | pdf |
    - Fig. 4.6.3.24. Pentagrammal relationships between scaling symmetry-related positive structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in parallel space (p. 509) | html | pdf |
    - Fig. 4.6.3.25. Pentagrammal relationships between scaling symmetry-related negative structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in parallel space (p. 509) | html | pdf |
    - Fig. 4.6.3.26. Pentagrammal relationships between scaling symmetry-related structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in parallel space (p. 509) | html | pdf |
    - Fig. 4.6.3.27. Pentagrammal relationships between scaling symmetry-related structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in perpendicular space (p. 510) | html | pdf |
    - Fig. 4.6.3.28. Perspective (a) parallel- and (b) perpendicular-space views of the reciprocal basis of the 3D Penrose tiling (p. 510) | html | pdf |
    - Fig. 4.6.3.29. Schematic representation of a rotation in 6D space (p. 511) | html | pdf |
    - Fig. 4.6.3.30. The two unit tiles of the 3D Penrose tiling: a prolate $[[\alpha_{p} = \hbox{arc} \cos (5^{-1/2}) \simeq 63.44^{\circ}]]$ and an oblate $[(\alpha_{o} = 180^{\circ} - \alpha_{p})]$ rhombohedron with equal edge lengths $[a_{r}]$ (p. 511) | html | pdf |
    - Fig. 4.6.3.31. Atomic surface of the 3D Penrose tiling in the 6D hypercubic description (p. 511) | html | pdf |
    - Fig. 4.6.3.32. Perspective parallel-space view of the two alternative reciprocal bases of the 3D Penrose tiling (p. 512) | html | pdf |
    - Fig. 4.6.3.33. Physical-space diffraction patterns of the 3D Penrose tiling decorated with point atoms (p. 513) | html | pdf |
    - Fig. 4.6.3.34. Perpendicular-space diffraction patterns of the 3D Penrose tiling decorated with point atoms (p. 514) | html | pdf |
    - Fig. 4.6.3.35. Radial distribution function of the structure factors $[F({\bf H})]$ of the 3D Penrose tiling (p. 515) | html | pdf |
    - Fig. 4.6.3.36. Parallel-space distribution of (a) positive and (b) negative structure factors of the 3D Penrose tiling of the 6D P lattice type decorated with point atoms (p. 515) | html | pdf |
    - Fig. 4.6.3.37. Perpendicular-space distribution of (a) positive and (b) negative structure factors of the 3D Penrose tiling of the 6D P lattice type decorated with point atoms (p. 516) | html | pdf |
    - Fig. 4.6.4.1. Simulated diffraction patterns of (a) the $[\simeq 52\;\hbox{\AA}]$ single-crystal approximant of decagonal Al–Co–Ni, (b) the fivefold twinned approximant, and (c) the decagonal phase itself (p. 517) | html | pdf |
    - Fig. 4.6.4.2. Zero-layer X-ray diffraction patterns of decaprismatic Al_73.5C_21.7Ni_4.8 crystals (p. 518) | html | pdf |
  - Tables
    - Table 4.6.2.1. Expansion of the Fibonacci sequence $[B_{n} = \sigma^{n}(\hbox{L})]$ by repeated action of the substitution rule σ: $[\hbox{S} \rightarrow \hbox{L}]$ , $[\hbox{L} \rightarrow \hbox{LS}]$ (p. 491) | html | pdf |
    - Table 4.6.3.1. 3D point groups of order k describing the diffraction symmetry and corresponding 5D decagonal space groups with reflection conditions (see Rabson et al., 1991) (p. 507) | html | pdf |
    - Table 4.6.3.2. 3D point groups of order k describing the diffraction symmetry and corresponding 6D decagonal space groups with reflection conditions (see Levitov & Rhyner, 1988; Rokhsar et al., 1988) (p. 514) | html | pdf |
    - Table 4.6.4.1. Intensity statistics of the Fibonacci chain for a total of 161 322 reflections with $[-1000 \leq h_{i} \leq 1000]$ and $[0 \leq \sin \theta/\lambda \leq 2\;\hbox{\AA}^{-1}]$ (p. 516) | html | pdf |

Dynamical theory and its applications
- 5.1. Dynamical theory of X-ray diffraction (pp. 534-551) | html | pdf | chapter contents |
  A. Authier
  - 5.1.1. Introduction (p. 534) | html | pdf |
  - 5.1.2. Fundamentals of plane-wave dynamical theory (pp. 534-538) | html | pdf |
    - 5.1.2.1. Propagation equation (pp. 534-535) | html | pdf |
    - 5.1.2.2. Wavefields (pp. 535-536) | html | pdf |
    - 5.1.2.3. Boundary conditions at the entrance surface (p. 536) | html | pdf |
    - 5.1.2.4. Fundamental equations of dynamical theory (p. 536) | html | pdf |
    - 5.1.2.5. Dispersion surface (pp. 536-537) | html | pdf |
    - 5.1.2.6. Propagation direction (pp. 537-538) | html | pdf |
  - 5.1.3. Solutions of plane-wave dynamical theory (pp. 538-541) | html | pdf |
    - 5.1.3.1. Departure from Bragg's law of the incident wave (p. 538) | html | pdf |
    - 5.1.3.2. Transmission and reflection geometries (pp. 538-539) | html | pdf |
    - 5.1.3.3. Middle of the reflection domain (p. 539) | html | pdf |
    - 5.1.3.4. Deviation parameter (p. 539) | html | pdf |
    - 5.1.3.5. Pendellösung and extinction distances (pp. 539-540) | html | pdf |
    - 5.1.3.6. Solution of the dynamical theory (p. 540) | html | pdf |
    - 5.1.3.7. Geometrical interpretation of the solution in the zero-absorption case (pp. 540-541) | html | pdf |
      - 5.1.3.7.1. Transmission geometry (p. 540) | html | pdf |
      - 5.1.3.7.2. Reflection geometry (p. 541) | html | pdf |
  - 5.1.4. Standing waves (p. 541) | html | pdf |
  - 5.1.5. Anomalous absorption (p. 541) | html | pdf |
  - 5.1.6. Intensities of plane waves in transmission geometry (pp. 541-545) | html | pdf |
    - 5.1.6.1. Absorption coefficient (pp. 541-542) | html | pdf |
    - 5.1.6.2. Boundary conditions for the amplitudes at the entrance surface – intensities of the reflected and refracted waves (p. 542) | html | pdf |
    - 5.1.6.3. Boundary conditions at the exit surface (pp. 542-543) | html | pdf |
      - 5.1.6.3.1. Wavevectors (pp. 542-543) | html | pdf |
      - 5.1.6.3.2. Amplitudes – Pendellösung (p. 543) | html | pdf |
    - 5.1.6.4. Reflecting power (pp. 543-544) | html | pdf |
    - 5.1.6.5. Integrated intensity (pp. 544-545) | html | pdf |
      - 5.1.6.5.1. Non-absorbing crystals (pp. 544-545) | html | pdf |
      - 5.1.6.5.2. Absorbing crystals (p. 545) | html | pdf |
    - 5.1.6.6. Thin crystals – comparison with geometrical theory (p. 545) | html | pdf |
  - 5.1.7. Intensity of plane waves in reflection geometry (pp. 545-548) | html | pdf |
    - 5.1.7.1. Thick crystals (pp. 545-546) | html | pdf |
      - 5.1.7.1.1. Non-absorbing crystals (pp. 545-546) | html | pdf |
      - 5.1.7.1.2. Absorbing crystals (p. 546) | html | pdf |
    - 5.1.7.2. Thin crystals (pp. 546-548) | html | pdf |
      - 5.1.7.2.1. Non-absorbing crystals (pp. 546-547) | html | pdf |
      - 5.1.7.2.2. Absorbing crystals (pp. 547-548) | html | pdf |
  - 5.1.8. Real waves (pp. 548-550) | html | pdf |
    - 5.1.8.1. Introduction (p. 548) | html | pdf |
    - 5.1.8.2. Borrmann triangle (pp. 548-549) | html | pdf |
    - 5.1.8.3. Spherical-wave Pendellösung (pp. 549-550) | html | pdf |
  - Appendix 5.1.1. (pp. 550-551) | html | pdf |
    - A5.1.1.1. Dielectric susceptibility – classical derivation (p. 550) | html | pdf |
    - A5.1.1.2. Maxwell's equations (pp. 550-551) | html | pdf |
    - A5.1.1.3. Propagation equation (p. 551) | html | pdf |
    - A5.1.1.4. Poynting vector (p. 551) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 5.1.2.1. Bragg reflection (p. 535) | html | pdf |
    - Fig. 5.1.2.2. Boundary condition for wavevectors at the entrance surface of the crystal (p. 536) | html | pdf |
    - Fig. 5.1.2.3. Intersection of the dispersion surface with the plane of incidence (p. 537) | html | pdf |
    - Fig. 5.1.2.4. Intersection of the dispersion surface with the plane of incidence shown in greater detail (p. 537) | html | pdf |
    - Fig. 5.1.2.5. Dispersion surface for the two states of polarization (p. 538) | html | pdf |
    - Fig. 5.1.3.1. Departure from Bragg's law of an incident wave (p. 538) | html | pdf |
    - Fig. 5.1.3.2. Transmission, or Laue, geometry (p. 538) | html | pdf |
    - Fig. 5.1.3.3. Reflection, or Bragg, geometry (p. 539) | html | pdf |
    - Fig. 5.1.3.4. Boundary conditions at the entrance surface for transmission geometry (p. 539) | html | pdf |
    - Fig. 5.1.3.5. Boundary conditions at the entrance surface for reflection geometry (p. 540) | html | pdf |
    - Fig. 5.1.6.1. Variation of the effective absorption with the deviation parameter in the transmission case for the 400 reflection of GaAs using Cu Kα radiation (p. 542) | html | pdf |
    - Fig. 5.1.6.2. Variation of the intensities of the reflected and refracted waves in an absorbing crystal for the 220 reflection of Si using Mo Kα radiation, t = 1 mm (μt = 1.42) (p. 542) | html | pdf |
    - Fig. 5.1.6.3. Boundary condition for the wavevectors at the exit surface (p. 543) | html | pdf |
    - Fig. 5.1.6.4. Decomposition of a wavefield into its two components when it reaches the exit surface (p. 543) | html | pdf |
    - Fig. 5.1.6.5. Cross sections of the incident, $[{\bf K}_{{\bf o}}^{(a)}]$ , refracted, $[{\bf K}_{{\bf o}}]$ , and reflected, $[{\bf K}_{{\bf h}}]$ , waves (p. 544) | html | pdf |
    - Fig. 5.1.6.6. Theoretical rocking curves in the transmission case for non-absorbing crystals and for various values of $[t/\Lambda_{L}]$ (p. 544) | html | pdf |
    - Fig. 5.1.6.7. Variations with crystal thickness of the integrated intensity in the transmission case (no absorption) (arbitrary units) (p. 545) | html | pdf |
    - Fig. 5.1.7.1. Theoretical rocking curve in the reflection case for a non-absorbing thick crystal in terms of the deviation parameter (p. 545) | html | pdf |
    - Fig. 5.1.7.2. Theoretical rocking curve in the reflection case for a thick absorbing crystal (p. 546) | html | pdf |
    - Fig. 5.1.7.3. Bragg case: thick crystals (p. 546) | html | pdf |
    - Fig. 5.1.7.4. Bragg case: thin crystals (p. 547) | html | pdf |
    - Fig. 5.1.8.1. Borrmann triangle (p. 548) | html | pdf |
    - Fig. 5.1.8.2. Packet of wavefields of divergence Δα excited in the crystal by an incident wavepacket of angular width $[\delta (\Delta \theta)]$ (p. 549) | html | pdf |
    - Fig. 5.1.8.3. Intensity distribution along the base of the Borrmann triangle (p. 549) | html | pdf |
    - Fig. 5.1.8.4. Interference at the origin of the Pendellösung fringes in the case of an incident spherical wave (p. 550) | html | pdf |
    - Fig. 5.1.8.5. Spherical-wave Pendellösung fringes observed on a wedge-shaped crystal (p. 550) | html | pdf |
- 5.2. Dynamical theory of electron diffraction (pp. 552-556) | html | pdf | chapter contents |
  A. F. Moodie, J. M. Cowley and P. Goodman
  - 5.2.1. Introduction (p. 552) | html | pdf |
  - 5.2.2. The defining equations (p. 552) | html | pdf |
  - 5.2.3. Forward scattering (p. 552) | html | pdf |
  - 5.2.4. Evolution operator (pp. 552-553) | html | pdf |
  - 5.2.5. Projection approximation – real-space solution (p. 553) | html | pdf |
  - 5.2.6. Semi-reciprocal space (p. 553) | html | pdf |
  - 5.2.7. Two-beam approximation (pp. 553-554) | html | pdf |
  - 5.2.8. Eigenvalue approach (p. 554) | html | pdf |
  - 5.2.9. Translational invariance (p. 554) | html | pdf |
  - 5.2.10. Bloch-wave formulations (p. 555) | html | pdf |
  - 5.2.11. Dispersion surfaces (p. 555) | html | pdf |
  - 5.2.12. Multislice (p. 555) | html | pdf |
  - 5.2.13. Born series (pp. 555-556) | html | pdf |
  - 5.2.14. Approximations (p. 556) | html | pdf |
  - References | html | pdf |
- 5.3. Dynamical theory of neutron diffraction (pp. 557-569) | html | pdf | chapter contents |
  M. Schlenker and J.-P. Guigay
  - 5.3.1. Introduction (p. 557) | html | pdf |
  - 5.3.2. Comparison between X-rays and neutrons with spin neglected (pp. 557-558) | html | pdf |
    - 5.3.2.1. The neutron and its interactions (p. 557) | html | pdf |
    - 5.3.2.2. Scattering lengths and refractive index (pp. 557-558) | html | pdf |
    - 5.3.2.3. Absorption (p. 558) | html | pdf |
    - 5.3.2.4. Differences between neutron and X-ray scattering (p. 558) | html | pdf |
    - 5.3.2.5. Translating X-ray dynamical theory into the neutron case (p. 558) | html | pdf |
  - 5.3.3. Neutron spin, and diffraction by perfect magnetic crystals (pp. 558-561) | html | pdf |
    - 5.3.3.1. Polarization of a neutron beam and the Larmor precession in a uniform magnetic field (pp. 558-559) | html | pdf |
    - 5.3.3.2. Magnetic scattering by a single ion having unpaired electrons (pp. 559-560) | html | pdf |
    - 5.3.3.3. Dynamical theory in the case of perfect ferromagnetic or collinear ferrimagnetic crystals (pp. 560-561) | html | pdf |
    - 5.3.3.4. The dynamical theory in the case of perfect collinear antiferromagnetic crystals (p. 561) | html | pdf |
    - 5.3.3.5. The flipping ratio (p. 561) | html | pdf |
  - 5.3.4. Extinction in neutron diffraction (non-magnetic case) (pp. 561-562) | html | pdf |
  - 5.3.5. Effect of external fields on neutron scattering by perfect crystals (p. 562) | html | pdf |
  - 5.3.6. Experimental tests of the dynamical theory of neutron scattering (pp. 562-563) | html | pdf |
  - 5.3.7. Applications of the dynamical theory of neutron scattering (pp. 563-564) | html | pdf |
    - 5.3.7.1. Neutron optics (p. 563) | html | pdf |
    - 5.3.7.2. Measurement of scattering lengths by Pendellösung effects (p. 563) | html | pdf |
    - 5.3.7.3. Neutron interferometry (pp. 563-564) | html | pdf |
    - 5.3.7.4. Neutron diffraction topography and other imaging methods (p. 564) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 5.3.3.1. Schematic plot of the two-beam dispersion surface in the case of a purely magnetic reflection such that $[{\bf Q}_{{\bf h}} = {\bf Q}_{-{\bf h}} = {\bf Q}_{{\bf 0}}]$ and that the angle between $[{\bf Q}_{{\bf 0}}]$ and $[{\bf Q}_{{\bf h}}]$ is equal to $[\pi /4]$ (p. 560) | html | pdf |

International Tables for CrystallographyVolume B: Reciprocal space

Edited by U. Shmueli

Contents

International Tables for Crystallography
Volume B: Reciprocal space